Top-degree components of Grothendieck and Lascoux polynomials

The Castelnuovo-Mumford polynomial $\widehat{\mathfrak{G}}_w$ with $w \in S_n$ is the highest homogeneous component of the Grothendieck polynomial $\mathfrak{G}_w$. Pechenik, Speyer and Weigandt define a statistic $\mathsf{rajcode}(\cdot)$ on $S_n$ that gives the leading monomial of $\widehat{\mathfrak{G}}_w$. We introduce a statistic $\mathsf{rajcode}(\cdot)$ on any diagram $D$ through a combinatorial construction ``snow diagram'' that augments and decorates $D$. When $D$ is the Rothe diagram of a permutation $w$, $\mathsf{rajcode}(D)$ agrees with the aforementioned $\mathsf{rajcode}(w)$. When $D$ is the key diagram of a weak composition $\alpha$, $\mathsf{rajcode}(D)$ yields the leading monomial of $\widehat{\mathfrak{L}}_\alpha$, the highest homogeneous component of the Lascoux polynomials $\mathfrak{L}_\alpha$. We use $\widehat{\mathfrak{L}}_\alpha$ to construct a basis of $\widehat{V}_n$, the span of $\widehat{\mathfrak{G}}_w$ with $w \in S_n$. Then we show $\widehat{V}_n$ gives a natural algebraic interpretation of a classical $q$-analogue of Bell numbers.


Introduction
Introduced by Lascoux and Schützenberger [LS82a], the Grothendieck polynomial G w is a polynomial representative of the K-class of structure sheaves of Schubert varieties of flag varieties.It is the inhomogeneous analogue of the Schubert polynomial S w : The lowest-degree component of G w forms S w .Pechenik, Speyer and Weigandt [PSW21] introduce the Castelnuovo-Mumford polynomial p G w 1 , the top-degree component of G w .They describe the leading monomial of p G w with respect to the tail lexicographic order by defining a new statistic rajcodep¨q on S n .We summarize some of their results on p G w .
(A) The polynomial p G w has leading monomial x rajcodepwq .(B) We have p G w is a scalar multiple of p G u if and only if rajcodepwq " rajcodepuq.(C) If w is inverse fireworks (see §5), then x rajcodepwq has coefficient 1 in p G w .Moreover, there exists exactly one u 1 P S n that is inverse fireworks such that rajcodepuq " rajcodepu 1 q.
Dreyer, Mészáros and St. Dizier [DMS22] provide an alternative proof of (A) via the climbing chain model for Grothendieck polynomials introduced by Lenart, Robinson, and Sottile [LRS06].Hafner [Haf22] provides an alternative proof of (A) for vexillary permutations via bumpless pipedreams.
Schubert polynomials are related to key polynomials κ α which are indexed by weak compositions.The key polynomials are the characters of Demazure modules [Dem74].Both Schubert and key polynomials can be defined recursively via the divided difference operators (see §2).In addition, Schubert polynomials expand positively into key polynomials [RS95].The key polynomials also have inhomogeneous analogues called Lascoux polynomials L α [Las03].Grothendieck polynomials and Lascoux polynomials are related: An expansion of Grothendieck polynomials into Lascoux polynomials was conjectured by Reiner and Yong [RY21] and proven by Shimozono and Yu [SY23].
Due to the connection between G w and L α , one would expect the top Lascoux polynomial p L α , the top-degree component of L α , to parallel p G w .We define a statistic rajcodep¨q on weak compositions and show in §4 that p L α enjoy properties analogous to the properties of p G w listed in Theorem 1.1: Theorem 1.2.Let α and γ be two weak compositions.
(a) The polynomial p L α has leading monomial x rajcodepαq .(b) We have p L α is a scalar multiple of p L γ if and only if rajcodepαq " rajcodepγq.(c) We say α is snowy if its positive entries are distinct.If α is snowy, then x rajcodepαq has coefficient 1 in p L α .Moreover, there exists exactly one snowy weak composition γ 1 such that rajcodepγq " rajcodepγ 1 q.
Our definition of rajcodep¨q on weak compositions is diagrammatic.Given a diagram D, we define a combinatorial construction called the snow diagram that augments and decorates D. Let rajcodepDq be the weight of the snow diagram.Every weak composition α is naturally associated with a diagram called the key diagram Dpαq (see Subsection 2.2).Then we define rajcodepαq :" rajcodepDpαqq.
Snow diagrams unify the computation of leading monomials in p G w and p L α .Each permutation w is also associated with a diagram called the Rothe diagram RDpwq.In §5, we show rajcodepwq " rajcodepRDpwqq.In other words, we give a diagrammatic way to compute rajcodepwq.
Finally, let p V n :" Q-spant p G w : w P S n u and p V :" Ť ně1 p V n .In Proposition 2.7, we show p V is a filtered algebra.Theorem 1.1 can be used to construct a basis of p V n and p V consisting of p G w .In particular, the dimension of p V n is B n , the n th Bell number.In §6, we use Theorem 1.2 to construct another basis consisting of p L α .This basis allows us to compute the Hilbert series of p V n and p V involving a q-analogue of B n .
The rest of the paper is organized as follows.In §2, we provide necessary background information and notation.In §3, we construct a snow diagram from any diagram and define statistics rajcodep¨q and rajp¨q on all diagrams.In §4, we prove Theorem 1.2.In §5, we show the statistics rajcodep¨q and rajp¨q on a Rothe diagram are equivalent to that defined in [PSW21].We also relate the snow diagram to two classical constructions: Schensted insertion and the shadow diagram.In §6, we derive the Hilbert series of p V n and p V .In §7, we present several open problems and future directions.
2. Background 2.1.Polynomials.We provide necessary background for Grothendieck polynomials and Lascoux polynomials.Then we introduce p G w and p L α which span the spaces p V n and p V .The Grothendieck polynomials G w P Z ě0 rx 1 , x 2 , . . .srβs were recursively defined by Lascoux and Schützenberger [LS82a].Let B i p¨q be the divided difference operators acting on the polynomial ring.
For each i, define B i pf q :" , where s i is the operator that swaps x i and x i`1 .Then for Let S `be the set of permutations of t1, 2, . . .u such that only finitely many numbers are permuted.Take w P S `and assume w only permutes numbers in rns.Let w 1 P S n be the restriction of w to rns and define G w as G w 1 .It is shown in [LS82a] that G w is well-defined.
A weak composition is an infinite sequence of non-negative integers with finitely many positive entries.Let C `be the set of weak compositions.For α P C `, we use α i to denote its i th entry, and write α " pα 1 , α 2 , . . ., α n q where α n is the last positive entry.We use x α to denote the monomial ř n iě1 α i .The Lascoux polynomials L α , indexed by weak compositions, are in Z ě0 rx 1 , x 2 , . . .srβs.By [Las03], they are defined recursively where π i is the operator π i pf q :" B i px i f q.
We say a pair pi, jq is an inversion of w P S n if i ă j and wpiq ą wpjq.Let Invpwq be the set of all inversions in w and let invpwq " |Invpwq|.Then we may view G w as a polynomial in β, where is a homogeneous polynomial in the x-variables with degree invpwq `d in Z ě0 rx 1 , x 2 , . . .s.The Schubert polynomial S w :" rβ 0 sG w .Similarly, viewing L α as a polynomial of β, rβ d sL α is a homogeneous polynomial with degree |α| `d in Z ě0 rx 1 , x 2 , . . .s.The key polynomial κ α :" rβ 0 sL α .The representation theoretic, geometric and combinatorial perspectives of Schubert polynomials and key polynomials are well-studied [Dem74,LS88,BJS93].
Define V n :" Q-spantS w : w P S n u and V :" Q-spantS w : In this paper, we are interested in the top-degree components of G w and L α .For a polynomial f P Qrx 1 , x 2 , . . .srβs, let p f " rβ d spf q where d is the largest such that rβ d spf q ‰ 0. The Castelnuovo-Mumford polynomial of w P S `is defined as p G w .The top Lascoux polynomial of α P C `is defined as p L α .In appendix §8, we list some Grothendieck polynomials and Lascoux polynomials.Pechenik, Speyer and Weigandt [PSW21] first study p G w .To the best of the authors knowledge, p L α has not been studied previously.Now consider the tail lexicographic order on monomials in the x-variables.We say a monomial x α is larger than x γ if there exists k such that α k ą γ k and α j " γ j for all j ą k.The leading monomial of f P Qrx 1 , x 2 , . . .s is the largest monomial in f .Among the four homogeneous polynomials above, three of them have combinatorial rules for their leading terms: (1) [BJS93] The leading monomial of S w with w P S n is x invcodepwq , where invcodepwq i " |tj : pi, jq P Invpwqu|.(2) [LS89] The leading monomial of κ α is x α .
(3) [PSW21] The leading monomial of p G w is x rajcodepwq defined as follows.
Definition 2.1.[PSW21] Let LIS w pqq be the length of the longest increasing subsequence of w P S n that starts with q.The rajcodepwq for w P S n is a weak composition where rajcodepwq r :" n `1 ŕ ´LIS w pwprqq for r P rns and 0 if r ą n.Then rajpwq :" |rajcodepwq|.
We will define rajcodep¨q on C `and show the leading monomial of L α is x rajcodepαq in §4.A connection between G w and L α is established by Shimozono and Yu [SY23].To describe this connection, we need the following notion.Example 2.8.The following are two examples of key diagrams.For clarity, we put an "i" on the left of row i and put a small dot in each cell.
RDpwq " We recall a combinatorial formula for Lascoux polynomials.To simplify our description, we introduce the following definition.Then a ghost diagram is a labeled diagram where cells can be labeled by X.We call cells labeled by X as "ghosts".For a ghost diagram D, its excess, denoted as expDq, is the number of ghosts in D. Next, we define a move on ghost diagrams.
We pick a cell pr, cq and move it up, subject to the following requirements.‚ The cell pr, cq must be the rightmost cell in row r. ‚ The cell pr, cq is not a ghost.
‚ The cell pr, cq is moved to the lowest empty spot above it.‚ The cell pr, cq may jump over other cells but cannot jump over any ghosts.After the move, we may or may not leave a ghost at pr, cq.When we leave a ghost, we refer this move as a ghost move.
For a weak composition α, a ghost diagram is called a K-Kohnert diagram of α if it can be obtained from Dpαq by K-Kohnert moves.Let KKDpαq be the set of all K-Kohnert diagrams of α.As proved in [PY22], K-Kohnert diagrams give a formula for Lascoux polynomials.This rule was first conjectured by Ross and Yong [RY15].Notice that our convention is different from [PY22]: row 1 is the top most row in this paper while it is the bottom most row in [PY22].

Snow diagrams
We associate each diagram with a labeled diagram called the snow diagram which allows us to define two statistics on diagrams.For each diagram D, we describe the following algorithm that outputs snowpDq.Cells in snowpDq can be labeled by ‚ or ˚. -Iterate through rows of D from bottom to top.
-In each row r of D, find the rightmost cell pr, cq with no ‚ in column c.If such an pr, cq exists, label it by ‚ and put a cell labeled by ˚in pr 1 , cq for r 1 P rr ´1s and pr 1 , cq R D. We call cells labeled by ‚ dark clouds and cells labeled by ˚snowflakes.
Example 3.1.The following is a diagram together with its snow diagram.
The positions of dark clouds will be important, so we make the following definition.A diagram is a non-attacking rook diagram if it has at most one cell in each row or column.Let Rook `be the family of all non-attacking rook diagrams.
Remark 3.4.We make the following observations about darkpDq.
‚ By construction, darkpDq P Rook `. ‚ Take pr, cq P D. If there are no r 1 ą r with pr 1 , cq P darkpDq and there are no c 1 ą c with pr, c 1 q P darkpDq, then pr, cq P darkpDq.
Finally, we associate two statistics to each diagram via its snow diagram.
Definition 3.5.Let D be a diagram.The rajcode of D, rajcodepDq, is the weak composition wtpsnowpDqq.Let rajpDq denote |rajcodepDq|, the total number of cells in snowpDq.
Remark 3.7.Recall that Pechenik, Speyer and Weigandt [PSW21] define the statistics rajcodep¨q and rajp¨q on permutations using increasing subsequences.We show that our rajcode and raj on Rothe diagrams agree with their definitions in Theorem 5.6.Therefore, our construction on Rothe diagrams is a diagrammatic way to compute the leading monomial and degree of p G w .In addition, we notice that positions of dark clouds in snowpRDpwqq are connected to the Schensted insertion and Viennot's geometric construction.These connections are explored in §5.

Proof of Theorem 1.2
To prove Theorem 1.2, we study top Lascoux polynomials via snow diagrams of key diagrams.With a slight abuse of notation, we define rajcodepαq :" rajcodepDpαqq, rajpαq :" rajpDpαqq and darkpαq " darkpDpαqq for α P C `.We start by introducing some definitions.
Definition 4.1.A weak composition α is called snowy if its positive entries are all distinct.
Our main goal in this section is to establish Theorem 1.2: Theorem 1.2.Let α and γ be two weak compositions.
(a) The polynomial p L α has leading monomial x rajcodepαq .(b) We have p L α is a scalar multiple of p L γ if and only if rajcodepαq " rajcodepγq.(c) We say α is snowy if its positive entries are distinct.If α is snowy, then x rajcodepαq has coefficient 1 in p L α .Moreover, there exists exactly one snowy weak composition γ 1 such that rajcodepγq " rajcodepγ 1 q.
This task is broken into four major lemmas established in the following four subsections.In Subsection 4.1, we use K-Kohnert diagrams to establish the first major lemma: Lemma 4.2.The polynomial L α has the term x rajcodepαq β rajpαq´|α| .Lemma 4.2 proves p L α has degree at least rajpαq.To show p L α indeed has degree rajpαq, we need the following equivalence relation on weak compositions.Definition 4.3.Let α and γ be two weak compositions.We say α is rajcode equivalent to γ, denoted as α " γ, if rajcodepαq " rajcodepγq.
In Subsection 4.2, we study this equivalence relation.We show that snowy weak compositions form a complete set of representatives: Lemma 4.5.For each equivalence class of ", there is a unique α such that α is snowy.Moreover, if γ " α and α is snowy, then γ r ě α r for all r.In other words, a snowy weak composition is the unique entry-wise minimum in each equivalence class.
In Subsection 4.3, we focus on p L α for snowy α and give a recursive description of p L α , which leads to the third major lemma.
Lemma 4.6.If α is snowy, then x rajcodepαq is the leading monomial of p L α with coefficient 1.
Finally, we devote the Subsection 4.4 to proving the last major lemma: Once we have these four major lemmas, we can easily check Theorem 1.2.
Proof.First, statement (c) follows from Lemma 4.5 and Lemma 4.6.Given a weak composition α.Let β be the unique snowy weak composition such that α " β.Statement (a) follows from Lemma 4.6 and Lemma 4.7.
For statement (b), the backward direction is just Lemma 4.7.For the forward direction, if p L α is a scalar multiple of p L γ , then they have the same leading monomial.By statement (a), we have rajcodepαq " rajcodepγq.
4.1.Proof of Lemma 4.2.We show the monomial x rajcodepαq β rajpαq´|α| exists in L α .We give an algorithm whose output is a K-Kohnert diagram for α, which has the same underlying diagram as snowpDpαqq.First, observe that snowpDpαqq contains no dark clouds if and only if α contains only zero entries.In this case, p L α " 1 and rajcodepαq only has zero entries.Our claim is immediate.In the rest of this subsection, we assume α is a weak composition with at least one positive entry, and thus snowpDpαqq has at least one dark cloud.To describe the algorithm, we introduce two useful moves on ghost diagrams.Definition 4.8.Let D be a ghost diagram.Let pr, cq be a non-ghost cell in D and let pr 1 , cq be the highest empty space in column c.If r 1 ă r, let U P pr,cq pDq be the diagram we get after moving pr, cq to pr 1 , cq.Let U P G pr,cq pDq be the diagram we get after moving pr, cq to pr 1 , cq and putting a ghost on pr, cq and all empty spaces between pr, cq and pr 1 , cq.If r 1 ą r, define U P G pr,cq pDq " U P pr,cq pDq " D.
Remark 4.9.Assume U P pr,cq or U P G pr,cq moves a cell to pr 1 , cq.Then this move can be achieved by a sequence of K-Kohnert moves if both of the following conditions hold for each r 1 ă j ď r: ‚ If pj, cq R D, then D has no cell to the right of column c in row j. ‚ If pj, cq P D, then it is not a ghost cell.Now we can describe the algorithm.Let D 0 " Dpαq.Recall by Remark 3.4, there is at most one dark cloud in each column of snowpDpαqq.We can label all the dark clouds as pr 1 , c 1 q, . . ., pr m , c m q where c 1 ă c 2 ¨¨¨ă c m for some m ě 1.We iterate i from 1 to m.At iteration i, compute Example 4.10.Consider α " p1, 3, 4, 0, 4, 3q, we compute its snow diagram and we have the dark clouds at p2, 1q, p3, 2q, p6, 3q, p5, 4q.We compute D 4 according to the above algorithm.
snowpDpαqq " We observe that in the previous example, D 4 has the same underlying diagram as snowpDpαqq.This is true in general.Proof.For a number c, we compare the column c of snowpDpαqq and D m .If column c of snowpDpαqq has no dark cloud, then it is the same as column c of Dpαq.In this case, the algorithm will not move any cells in column c.Thus, D m and Dpαq also agree in column c.
Now suppose snowpDpαqq has a dark cloud in column c, say at row r.In the underlying diagram of snowpDpαqq, column c is obtained from column c of Dpαq by filling all empty spaces above row r.On the other hand, consider what the algorithm does on column c.It first might move cells above row r and then it fills all empty spaces weakly above row r.Thus, column c in D m is the same as column c of snowpDpαqq after ignoring the labels.
Next, we want to show D m produced by the algorithm is in KKDpαq.We just need to check each U P pr,jq and U P G pr,cq in each iteration is a sequence of K-Kohnert moves.To that end, we first make the following observation about the diagram D i .Lemma 4.12.Let c 0 " 0. In D i , if a cell is strictly to the right of column c i , then there is a cell immediately on its left.In other words, the diagram D i is left-justified if we ignore the first c i columns.
Proof.Prove by induction on i.The lemma holds for D 0 , which is left-justified.
Assume D i´1 is left-justified if we ignore the first c i´1 columns, for some i ě 1.Consider an arbitrary cell pr, cq in D i with c ą c i .We show pr, c ´1q is in D i by considering two possibilities.
-The cell pr, cq is not in D i´1 .Then during iteration i, a cell is moved to pr, cq, which is the highest blank in column c of D i´1 .By our inductive hypothesis and c ´1 ą c i´1 , the highest blank in column c ´1 of D i´1 is weakly lower than row r.Thus, pr, c ´1q is in D i .-Otherwise, pr, cq is in D i´1 .By our inductive hypothesis, pr, c ´1q is in D i´1 .If r ‰ r i , then we know that no cell from row r is moved during iteration i.Thus, pr, c ´1q is still in D i .If r " r i , then there are no empty spaces above pr, cq in D i´1 .By our inductive hypothesis, there is no empty spaces above pr, c ´1q, so pr, c ´1q is still in D i .
The above lemma shows that the diagram D i is left-justified if we ignore the first c i columns.We will use this property to show that D m is in KKDpαq.
Proposition 4.13.The above algorithm can be achieved by K-Kohnert moves, so D m P KKDpαq.
Proof.We focus on one iteration of the algorithm, say iteration i.We check the operators in (1) can be achieved by K-Kohnert moves.We ignore all cells to the left of the column c i in D i´1 .By the previous Lemma, this part of the diagram is left-justified.The highest empty spaces in columns c i , ¨¨¨, α r i are going weakly up from left to right.Moreover, the condition in Remark 4.9 holds for all pr i , c i q, ¨¨¨, pr i , α r i q.
Now U P pr i ,αr i q can be achieved by K-Kohnert moves.After that, the conditions in Remark 4.9 hold at each step for pr i , α r i ´1q, . . ., pr i , c i q.Following this logic, this iteration can be achieved by K-Kohnert moves.
Using Theorem 2.12: Lemma 4.2.The polynomial L α has the term x rajcodepαq β rajpαq´|α| .4.2.Proof of Lemma 4.5.First, notice that we can recover the underlying diagram of snowpDpαqq from darkpαq.Proof.First, we show that the elements of the set (2) are cells in snowpDpαqq.Take pr, cq P darkpαq.We know pr, cq P Dpαq.Since Dpαq is left-justified, tru ˆrcs Ď Dpαq.Thus, these cells are in snowpDpαqq.By the construction of snowpDpαqq, the cells in rrs ˆtcu are also in snowpDpαqq.
Now suppose there is a cell pr, cq in snowpDpαqq that is not in the set (2).Then there is no r 1 ą r with pr 1 , cq P darkpDq, which implies pr, cq is not a snowflake in snowpDpαqq.Thus, pr, cq P Dpαq.Also, there is no c 1 ą c with pr, c 1 q P darkpDq.By Remark 3.4, pr, cq P darkpDq.Thus, pr, cq is in the set (2), which is a contradiction.Furthermore, we can recover darkpαq from rajcodepαq.
Proof.We prove the two diagrams darkpαq and darkpγq agree on each row r, by a reverse induction on r.The base case is immediate.Suppose r is large enough such that α i " γ i " 0 if i ą r.Then darkpαq and darkpγq clearly agree on row r and underneath.
Next, we show that the value rajcodepαq r and cells in darkpαq under row r determines whether darkpαq has a cell on row r.Moreover, if such a cell exists, its column index is also determined.
Let r ě 1. Define B r :" tc : There are no dark clouds under pr, cq in snowpDpαqu.
The complement of B r is B r :" Z ą0 ´Br " tc : pr 1 , cq P darkpαq for some r 1 ą ru.For c P B r , pr, cq of snowpDpαqq is a snowflake or an unlabeled cell.If there is no dark cloud on row r of snowpDpαqq, rajcodepαq r " |B r |.Otherwise, we assume the dark cloud is at pr, cq for some c P B r .Then row r of snowpDpαqq has cells on pr, c 1 q for c 1 P B r or c 1 ď c.Suppose c is the i th smallest number in B r .We have rajcodepαq r " i `|B r |.
Consequently, rajcodepαq r and darkpαq under row r uniquely determines row r of darkpαq.If we assume darkpαq and darkpγq agree underneath row r as our inductive hypothesis, then they also agree on row r since rajcodepαq r " rajcodepγq r .The induction is finished.Now we have two equivalent ways of describing rajcode equivalence.
Our next goal is to find representatives of rajcode equivalence classes.At the end of this subsection, we will see snowy weak compositions form a complete set of representatives.To understand snowy weak compositions, we start with the following observation.
Remark 4.17.For a weak composition α, the following are equivalent: ‚ α is snowy.‚ The rightmost cell in each row of Dpαq are in different columns.‚ The rightmost cell in each row of Dpαq is a dark cloud in snowpDpαqq.
One advantage of working with snowy weak compositions is that we can tell their rajcodep¨q and rajp¨q easily: Lemma 4.18.Let α be a snowy weak composition.Then the following statements hold.
As a consequence, we have the following rule which tells us how rajcodeps i αq differs from rajcodepαq when α is snowy.
Corollary 4.19.Let α be a snowy weak composition and consider i with α i ą α i`1 .Then rajcodeps i αq " s i rajcodepαq `ei , where e i is the weak composition with 1 on its i th entry and 0 elsewhere.
The second advantage of working with snowy weak compositions is that they are in bijection with Rook `.
Lemma 4.20.The map darkp¨q is a bijection from tα P C `: α is snowyu to Rook `.Its inverse dark ´1p¨q is given by dark ´1pRq " α where Proof.Follows from Remark 4.17.
We are ready to show that they are representatives of all equivalence classes.
Lemma 4.5.For each equivalence class of ", there is a unique α such that α is snowy.Moreover, if γ " α and α is snowy, then γ r ě α r for all r.In other words, a snowy weak composition is the unique entry-wise minimum in each equivalence class.
Proof.Let γ be an arbitrary weak composition.First, we construct a snowy α such that α " γ.
Finally, we establish the uniqueness of this snowy α.Assume α 1 is a snowy weak composition such that α 1 " γ.Then α 1 r ě α r and α r ě α 1 r for all r P Z ą0 , so α " α 1 .

4.3.
Proof of Lemma 4.6.By Lemma 4.2, p L α has degree at least rajpαq.Next, we can show the degree of p L α equals to rajpαq when α is snowy.
Lemma 4.21.Let α be a snowy weak composition.The β-degree of L α is rajpαq ´|α|, so the degree of p L α is rajpαq.
The β-degree in L α is at most rajpαq ´|α|.Lemma 4.2 implies the β-degree of L α is at least rajpαq ´|α|, so the inductive step is finished.

Now we can describe p
L α for snowy α recursively.
Lemma 4.23.Let α be a snowy weak composition.Then Proof.When α is weakly decreasing, our rule is immediate.Now assume α i ă α i`1 for some i P Z ą0 .By Corollary 4.19, rajps i αq " rajpαq ´1.We write L s i α as g `βrajpαq´1´|α| p L s i α for some g P Zrx 1 , x 2 , ¨¨¨srβs with β-degree less than rajpαq ´1 ´|α|.Now we write L α as L α " π i pL s i α q `βπ i px i`1 L s i α q " π i pL s i α q `βπ i px i`1 gq `βrajpαq´|α| π i px i`1 p L s i α q When we extract the coefficient of β rajpαq´|α| , the left-hand side is p L α .On the right-hand side, the first two terms are ignored and we get π i px i`1 p L s i α q.
Combining Lemma 4.2 and Lemma 4.21, we know x rajcodepαq appears in p L α when α is snowy.Next, we show this monomial is the leading monomial of p L α .We start with the following observation about the operator f Þ Ñ π i px i`1 f q.
Remark 4.24.Let γ be a monomial.We may describe the leading monomial of π i px i`1 x γ q and its coefficient as follows.
‚ If γ i ą γ i`1 , then x i x s i γ is the leading monomial with coefficient 1.
We can understand how the operator f Þ Ñ π i px i`1 f q changes the leading monomial of polynomial f satisfying certain conditions.Lemma 4.25.Take f P Zrx 1 , x 2 , ¨¨¨s with f ‰ 0. Assume x α is the leading monomial in f with coefficient c ‰ 0. Pick an i P Z ą0 such that α i ą α i`1 .Furthermore, assume for any monomial in f , its power of x i is at most α i .Then x i x s i α is the leading monomial in π i px i`1 f q with coefficient c.
Proof.In this proof, we use "ě" to denote the monomial order.Let Γ be the set of weak compositions γ such that x γ appears in f .Let c γ be the coefficient of x γ in f .We may write f " ř γPΓ c γ x γ .Then π i px i`1 f q " ř γPΓ c γ π i px i`1 x γ q.By the remark above, x i x s i α appears in c α π i px i`1 x α q as the leading monomial with coefficient c α " c.It is enough to show the following claim.Claim: Take γ P Γ such that π i px i`1 x γ q ‰ 0 (i.e.γ i ‰ γ i`1 ).Let x γ 1 be the leading monomial in π i px i`1 x γ q.If x γ 1 ě x i x s i α , then γ " α.Proof: Assume α ‰ γ.Let k be the largest index such that the power of x k differs in x γ 1 and x i x s i α .By x γ 1 ě x i x s i α , the power of x k in x γ 1 is greater than the power of x k in x i x s i α .We must have k ď i `1.Otherwise, x γ ą x α , which contradicts x α being the leading monomial in f .Now we know γ 1 , α and γ all agree after the pi `1q th entry.Then γ 1 i`1 is at least the power of x i`1 in x i x s i α , which is α i .On the other hand, by x γ ď x α , γ i`1 ď α i`1 .Thus, (4) , which is impossible.Thus, we must have γ i ą γ i`1 .By Remark 4.24 again, γ 1 i`1 " γ i .By the assumptions in the statement of the lemma, γ i ď α i , so Next, γ 1 i is at least the power of x i in x i x s i α , which is α i`1 `1.Remark 4.24 implies γ 1 i " γ i`1 `1.Thus, γ i`1 ě α i`1 .By (4), γ i`1 " α i`1 .Now we know k ă i and γ j " α j for j " i or i `1.Thus, γ j " α j for all j ą k, so x γ ą x α , which is a contradiction.Now we can establish our third major lemma.
Lemma 4.6.If α is snowy, then x rajcodepαq is the leading monomial of p L α with coefficient 1.
If ℓpαq " 0, then α is weakly decreasing, then L α " x α " x rajcodepαq .Our claim is immediate.Now if ℓpαq ą 0, we can find r with α r ă α r`1 .Pick the largest such r.For our inductive hypothesis, assume x rajcodepsrαq is the leading monomial of p L srα with coefficient 1.By the maximality of r, α r`1 ě α r`2 ě α r`3 ě ¨¨¨.Thus, in any K-Kohnert diagram of s r α, there cannot be more than α r`1 cells in row r.In other words, for any monomial of p L srα , the power of x r is at most α r`1 .Lemma 4.25 implies that x r x srrajcodepsrαq is the leading monomial of p L α with coefficient 1.Finally, by Corollary 4.19, x r x srrajcodepsrαq " x rajcodepαq .4.4.Proof of Lemma 4.7.We first derive two consequences of α " γ.We start with the following definition.
In plain words, D is the diagram obtained by filling the empty spaces above each cell of D. Then Dpαq is completely determined by darkpαq: Lemma 4.27.Let α be a weak composition.Then Dpαq " Ť pr,cqPdarkpαq rrs ˆrcs.Proof.We show each side is a subset of the other.Take pr 1 , c 1 q P Dpαq.By Remark 3.4, there is pr 2 , c 2 q P darkpαq such that r 2 ě r 1 and c 2 ě c 1 .Thus, rr 1 s ˆtc 1 u Ď rr 2 s ˆrc 2 s.
We have the following consequence of α " γ.
Another nice consequence of α " γ one might expect is s r α " s r γ.Unfortunately, this is not always true.It is easy to check p0, 1q " p1, 1q but s 1 p0, 1q " p1, 0q and s 1 p1, 1q " p1, 1q are not similar.However, it is true when α and r satisfy the following condition.
Lemma 4.29.Let α be a weak composition and r P Z ą0 .Assume there exists c such that pr, cq R snowpDpαqq but pr `1, cq P snowpDpαqq.Then (i) α r`1 ą α r ; (ii) The diagram darkps r αq is obtained from darkpαq by switching row r and row r `1; (iii) For any γ with γ " α, we must have γ r`1 ą γ r and s r α " s r γ.
Proof.Since pr, cq is not in snowpDpαqq, we can deduce two facts: (1) There are no dark clouds under row r in column c, and (2) α r ă c.By (1), the cell pr `1, cq in snowpDpαqq is not a dark cloud or a snowflake.Thus, it is unlabeled and pr `1, cq P Dpαq.By Remark 3.4, there must be a c 1 ą c such that pr `1, c 1 q is a dark cloud in snowpDpαqq.This implies α r`1 ą c.By (2), we have α r`1 ą α r , proving (i).Also by (2), the dark cloud in row r of snowpDpαqq, if exists, is in the first c ´1 columns.Thus, darkps r αq is obtained from darkpαq by switching row r and row r `1, proving (ii).Now consider any γ " α.By Proposition 4.16, snowpDpγqq and snowpDpαqq have the same underlying diagram.By (ii), darkps r γq is obtained from darkpγq by switching row r and row r `1.Since darkpαq " darkpγq, we have darkps r αq " darkps r γq, so s r α " s r γ.
These two consequences of α " γ allow us to prove the last main Lemma.
Proof.By Lemma 4.5 it is enough to assume γ is snowy, and we proceed by induction on rajpαq.The base case is rajpαq " 0, which implies α only has 0s.Our claim is immediate.Now assume rajpαq ą 0. Consider the diagram Dpαq.Clearly, the underlying diagram of any K-Kohnert diagram of α will be a subset of Dpαq.In other words, any monomial in p L α must divide x wtpDpαqq .
If the underlying diagram of snowpDpαqq is Dpαq, then x wtpDpαqq is the only monomial in p L α .On the other hand, Corollary 4.28 gives Dpαq " Dpγq.By the same argument, x wtpDpαqq is the only monomial in p L γ .Our claim holds.Otherwise, we can find pr, cq P Dpαq but not in snowpDpαqq.Choose the pr, cq with the largest r.First, we know pr, cq R Dpαq, which implies pr `1, cq P Dpαq.By the maximality of r, pr `1, cq is in snowpDpαqq.We invoke Lemma 4.29 and conclude α r`1 ą α r , γ r`1 ą γ r and s i α " s i γ.Since γ is snowy, by Corollary 4.19, we know rajps r γq " rajpγq ´1, which implies rajps r αq " rajpαq ´1.By our inductive hypothesis, p L srα " c p L srγ for some c ‰ 0. We may write L srα as β rajpsrαq´|α| p L srα `g, where g has β-degree less than rajps r αq ´|α|.Then The first two terms on the right-hand side have β degree less than rajpαq ´|α|.Thus, the βdegree in L α is at most rajpαq ´|α|.By Lemma 4.2, the β-degree in L α is rajpαq ´|α|.Extract the coefficient of β rajpαq´|α| and get p L α " π i px i`1 p L srα q " cπ i px i`1 p L srγ q " c p L γ , by Lemma 4.23.

Snow diagrams for Rothe diagrams
Fix an n P Z ą0 throughout this section.We move on to study the snow diagrams of RDpwq for w P S n .In subsection 5.1, we recall a version of Schensted insertion on S n .In subection 5.2, we show the positions of dark clouds in snowpRDpwqq is related to the Schested insertion.We then use this connection to prove that rajcodepRDpwqq is consistent to the rajcodepwq defined in [PSW21].In Section 5.3, we show the dark clouds in snowpRDpwqq corresponds to the turning points in the shadow diagram for w.In Section 5.4, we study the snow diagrams for inverse fireworks permutations.
5.1.The Schensted Insertion.If a diagram is top-justified and left-justified, we say it is a Young diagram.A filling of a Young diagram with positive integers is called a tableau.A tableau is called partial if it contains distinct numbers and each row (resp.column) is decreasing from left to right (resp.top to bottom).Notice that usually in literature, columns and rows are increasing.We reverse the convention to make our results easier to state.
The Schensted insertion [Sch61] is an algorithm defined on a partial tableau T and a positive number x that is not in T .It finds the largest x 1 in the first row of T such that x ą x 1 .
‚ If such x 1 does not exist, it appends x at the end of row one and terminates.‚ Otherwise, it replaces x 1 by x and insert x 1 to the next row in the same way.
When the algorithm terminates, the resulting partial tableau is the output.For w P S n , we insert wpnq, wpn ´1q, . . ., wp1q to the empty tableau via the Schensted insertion and denote the result by P pwq. .
One classical application of the Schensted insertion is to study increasing subsequences in a permutation.Recall LIS w pqq is the length of the longest increasing subsequence of w P S n that starts with q.It is related to the Schensted insertion as follows.
Lemma 5.2.[Sag01, Lemma 3.3.3]Take w P S n and perform the Schensted insertion on w.For any r P rns, when wprq is inserted, it goes to column LIS w pwprqq in row one.
Example 5.3.Consider the w P S 7 in Example 5.1.Notice that LIS w pwp4qq " 3. When wp4q " 1 is inserted to row one, it indeed goes to column 3. 5.2.Rajcode of Rothe diagrams.We show that rajcodepwq defined by Pechenik, Speyer and Weigandt (see Definition 2.1) agrees with the rajcodepRDpwqq (see Definition 3.5).To do so, we need a better understanding of snowpRDpwqq.We start by describing how the positions of dark clouds in snowpRDpwqq are related to the Schensted insertion described in Subsection 5.1.
Proposition 5.4.Take w P S n .Consider the Schensted insertion on w.The dark cloud in row r of snowpRDpwqq can be described based on the insertion of wprq.
(1) If wprq is appended to the end of row one, then there is no dark cloud in the r th row of snowpRDpwqq; (2) If wprq bumps c in row one, then pr, cq is a dark cloud in snowpRDpwqq.For two cells pi, jq, pm, nq P NˆN, pm, nq lies in the shadow of pi, jq if and only if m ď i and n ď j.This can be visualized by imagining shedding light from the Southeast.2To obtain the shadow diagram of w P S n , consider the points p1, wp1qq, . . ., pn, wpnqq.Let ´ip1q 1 , wpi p1q 1 q ¯, . . ., ´ip1q ℓ 1 , wpi p1q ℓ 1 q be the points that are not in the shadow of any other point for some ℓ 1 ě 1 and i p1q 1 ą i p1q 2 ą ¨¨¨ą i p1q ℓ 1 .Then the first shadow line L 1 pwq is the boundary of the combined shadows of the points ´ip1q 1 , wpi p1q 1 q ¯, . . ., ´ip1q ℓ 1 , wpi p1q ℓ 1 q ¯.The rest of the L j pwq can be constructed recursively.Supposed L 1 , . . ., L j´1 have been constructed, remove all points in the set !´i ppq k , wpi then L j is the boundary of the shadow of the remaining points of the points left, which we label as ´ipjq 1 , wpi pjq 1 q ¯, . . .
for some ℓ j ě 1 and i pjq Once there is no point left, the shadow lines we obtained form the shadow diagram for w.
Theorem 5.7 ([Vie77]).Given w P S n and suppose L 1 , . . ., L s are the shadow lines obtained from w until there is no point left.Then s equals the size of Row 1 pP pwqq.
In total, there are n ´|Row 1 pP pwqq| turning points for each w P S n .There is a classical result connecting these turning points to the Schensted insertion.Example 5.10.Consider w " 3721564 P S 7 .We present its Rothe diagram, its shadow diagram, and the snow diagram of RDpwq.From Example 5.1, the Schensted insertion on w yields a tableau whose row 1 has three cells.Correspondingly, there are three shadow lines.The turning points of the shadow lines are p3, 1q, p1, 2q, p6, 4q, p2, 6q, which are positions for dark clouds in snowpRDpwqq.
Remark 5.11.A geometric interpretation for the rajcode is given in [PSW21, Section 4] in terms of the "blob diagrams."Specifically, the set of points in the same shadow line in the shadow line diagram is labeled as B n , B n´1 , . . .from southeast to northwest.With the labeling on the blob diagrams, we can obtain the rajcode directly.That is, if pi, wpiqq P B k , then rajcodepwq i " k ´i.Definition 5.12.[PSW21, Definition 3.5] A permutation w P S n is a fireworks permutation if its initial element in each decreasing run is increasing.A permutation w P S n is an inverse fireworks permutation if w ´1 is a fireworks permutation.
Inverse fireworks permutations are the representatives of equivalence classes, given by permutations with the same rajcode [PSW21].The snowy weak compositions play the same role in our study of p L. We investigate the similarities between inverse fireworks permutations and snowy weak compositions.For w inverse fireworks, RDpwq enjoy analogous properties as the Dpαq of snowy α.We start with the following observation about RDpwq.
Lemma 5.13.Let w P S n be an inverse fireworks permutation.Consider each r P rns such that row r of RDpwq is not empty.The rightmost cell in row r of RDpwq is pr, wprq ´1q.
Proof.Recall that pr, wpr 1 qq P RDpwq if and only if pr, r 1 q P Invpwq if and only if pwpr 1 q, wprqq P Invpw ´1q.Let c " wprq.Clearly, cells in row r of RDpwq are within the first c ´1 columns.It remains to check pr, c ´1q P RDpwq, which is equivalent to pc ´1, cq P Invpw ´1q.
Since row r of RDpwq is nonempty, it must contain a cell pr, iq such that pi, cq P Invpw ´1q for some i P rc ´1s.Since w ´1piq ą w ´1pcq and w ´1 is fireworks, w ´1pcq can not be the initial element in its decreasing run.Therefore w ´1pc ´1q ą w ´1pcq and we have pc ´1, cq P Invpw ´1q.
We can characterize the inverse fireworks permutations using Rothe diagrams or the snow diagram of the permutation.This is similar to Remark 4.17, where we describe snowy weak compositions using key diagrams and dark clouds.
Proposition 5.14.Take w P S n .The following are equivalent: (1) w is an inverse fireworks permutation.
(2) In RDpwq, the rightmost cells in each row are in different columns.
(3) In snowpRDpwqq, the rightmost cell in each row is a dark cloud.
Proof.The last two statements are clearly equivalent.Now we establish the equivalence of the first two statements.
Assume w is inverse fireworks.Take r, r 1 P rns with r ‰ r 1 such that row r and row r 1 of RDpwq are not empty.By Lemma 5.13, the rightmost cell in row r (resp.r 1 ) is at pr, wprq ´1q (resp.pr 1 , wpr 1 q ´1q).Clearly, wprq ´1 ‰ wpr 1 q ´1, so we have our second statement.Now we assume w is not inverse fireworks.We can find a number r in w ´1 such that r is the initial element in its decreasing run, but r is less than r 1 , the initial element of the previous decreasing run.Let c 1 " wpr 1 q and c " wprq.Since pc 1 , cq P Invpw ´1q, pr, c 1 q P RDpwq.Thus, row r of RDpwq is not empty.Let pr, iq be the rightmost cell in row r.In other words, i is the largest such that pi, cq P Invpw ´1q.We have c 1 ď i ă c ´1.Consider the decreasing run before w ´1pcq: w ´1pc 1 q ą w ´1pc 1 `1q ą ¨¨¨ą w ´1pc ´1q.We see pi, i `1q is also in Invpw ´1q.In row w ´1pi `1q, the cell pw ´1pi `1q, iq is the rightmost cell of its row.Thus, the second statement does not hold, and the proof is finished.
With the above proposition, we can compute rajcodepwq easily if w is inverse fireworks.The following rule is similar to Lemma 4.18(2).
Proposition 5.15.Assume w P S n is inverse fireworks.For each r P rns, rajcodepwq r " |tr 1 ą r :pr, r 1 q P Invpwq or wpr 1 q ą wprq and pr 1 , r 2 q P Invpwq for some r 2 u|.
Proof.First, we know rajcodepwq r " rajcodepRDpwqq r is the number of cells in the r th row of snowpRDpwqq.The number of non-snowflake cells on this row is given by |tr 1 : pr, r 1 q P Invpwqu|.Now we count the number of snowflakes in row r of snowpRDpwqq.It is the number of r 1 ą r such that row r 1 of snowpRDpwqq has a dark cloud on the right of the column wprq.By Lemma 5.13, row r 1 has a dark cloud at column wpr 1 q ´1 if RDpwq is nonempty in row r 1 .Thus, the number of snowflakes in row r of snowpRDpwqq is the number of r 1 ą r such that wpr 1 q ą wprq and pr 1 , r 2 q P Invpwq for some r 2 .

Vector space spanned by p G w
We now study the vector spaces p V n :" Q-spant p G w : w P S n u and p V :" Q-spant p G w : w P S `u.By Theorem 1.1, they have bases t p G w : w P S n is inverse fireworksu and t p G w : w P S `is inverse fireworksu respectively.By [Cla01], the number of inverse fireworks permutations in S n is B n , the n th Bell number.Thus, p V n has dimension B n .We introduce another basis of p V n and p V consisting of p L α , the top-degree components of Lascoux polynomials.One application of the top Lascoux basis is to compute the Hilbert series of p V n and p V .For a vector space V Ď Qrx 1 , x 2 , ¨¨¨s, the Hilbert series of V is HilbpV ; qq :" where m d is the number of polynomials with degree d in a homogeneous basis of V .In Subsection 6.1, we recall the definition of B n and its q-analogue B n pqq.In Subsection 6.2, we compute Hilbp p V n ; qq using B n pqq and rook-theoretic results.In Subsection 6.3, we compute Hilbp p V ; qq.
6.1.Stirling numbers, Bell numbers and their q-analogues.Let n, k be non-negative integers throughout this subsection.Let S n,k be the Stirling number of the second kind, defined by the recurrence relation S n`1,k " S n,k´1 `kS n,k ,

Open Problems and Future Directions
We conclude with several open problems for future study.In Section 5.3, we present the connections between the following three constructions: -Positions of dark clouds in snowpRDpwqq; -First step of Viennot's geometric construction; -Bumps in the first row during Schensted insertion.
The Grothendieck to Lascoux expansion, proven in [SY23], involves finding certain tableaux and computing their right keys.G w pxq, we can think they correspond to a subset of pipe dreams for G w pxq.In [PSW21], the authors proved a factorization of x G w px, yq into a x-polynomial and a y-polynomial, and they showed the the leading term is in fact, x rajcodepwq y rajcodepw ´1q , with coefficient 1 by constructing a pipe dream associated with it iteratively.Problem 7.5.Use the snow diagrams to give an explicit construction of pipe dreams for the leading term in x G w px, yq.
Problem 7.6.Find characterizations of diagram D such that the leading monomial of p κ D is given by rajcodepDq.

Definition 2 .
10.A labeled diagram is a diagram where each cell can be labeled by a symbol.The underlying diagram of a labeled diagram is the diagram obtained by ignoring all labels.The weight of a labeled diagram D, denoted as wtpDq, is just the weight of its underlying diagram.
Lemma 4.11.The labeled diagram D m defined by (1) has the same underlying diagram as snowpDpαqq.

Lemma 4. 14 .
Let α be a weak composition.The underlying diagram of snowpDpαqq is: ď pr,cqPdarkpαq prrs ˆtcuq Y ptru ˆrcsq.(2) Example 5.1.Take w P S 7 with one-line notation 3721564.The Schensted insertion on w yields: Example 5.5.Let w P S 7 with one-line notation 3721564.Consider the corresponding Rothe diagram RDp3721564q and its snow diagram: of w is presented in Example 5.

j , wpi pjq ℓ j q for some ℓ j ě 1 and i pjq 1 ą i pjq 2
Theorem 5.8 ([Vie77,Knu70]).Let a shadow line L j of a permutation w consists of points ´ipjq 1 , wpi pjq 1 q ¯, . ..´ipjq ℓ ą ¨¨¨ą i pjq ℓ j .Then during Schensted insertion on w, when we insert wpi pjq k`1 q, it bumps wpi pjq k q from the first row.Combining Proposition 5.4 and Theorem 5.8, we have the following.Corollary 5.9.Each of the turning points in the shadow diagram of w contains a dark cloud in snowpwq.Any dark cloud in snowpwq is also a turning point in the shadow diagram of w.

Figure 1 .
Figure 1.Left: RD(w); Middle: shadow diagram of w; Right: snowpRDpwqq Problem 7.3.Find a combinatorial formula for the expansion of Castelnuovo-Mumford polynomials into top Lascoux polynomials indexed by snowy weak compositions.Finding a combinatorial formula for the structure constants c w u,v for Grothendieck polynomials, defined asG u G v " ÿ w c w u,v G w ,has been a long-standing open problem.These coefficients have a geometric interpretation: They are the intersection numbers for the Schubert classes in the connective K-theory.If we consider only the top-degree terms on both sides, we get the structure constants for Castelnuovo-Mumford polynomials, which we denote as x c w uv , which are still non-negative integers.Problem 7.4.Find a combinatorial formula for x c w uv .The Grothendieck polynomials G w pxq are a specialization of the double Grothendieck polynomials G w px, yq by setting y 1 " y 2 " ¨¨¨" 0. In[KM01], Knutson and Miller introduced pipe dream rules for both G w pxq and G w px, yq.For Castelnuovo-Mumford polynomials x are polynomials in Z ě0 rβsrx 1 , x 2 , ...s.We say f expands positively into tf 1 , f 2 , ...u if there exist g 1 , g 2 , ¨¨¨P Z ě0 rβs such that f " ř i g i f i .Theorem 2.4 ([SY23]).For w P S `, G w expands positively into tL α :α P C `u.Let f, f 1 , f 2 , . . . in Z ě0 rβsrx 1 , x 2 , ...s.If f expands positively into tf 1 , f 2 , ...u, then p f expands positively into p f 1 , p f 2 , . ...Bri02], the product G u G v with u P S m and v P S n expands positively into G w with w P S m`n .By Lemma 2.5, p G u p G v with u P S m and v P S n expands positively into p G w with w P S m`n .Finally, we conclude the following.Diagrams.A diagram is a finite subset of Z ą0 ˆZą0 .We represent a diagram by putting a cell at row r and column c for each pr, cq in the diagram.The leftmost column (resp.topmostrow) is called column 1 (resp.row1).The weight of a diagram D, denoted as wtpDq, is a weak composition whose i th entry is the number of boxes in its row i.We recall two classical families of diagrams.Each weak composition α is associated with a diagram called the key diagram, denoted as Dpαq.It is the unique left-justified diagram with weight α.One important key diagram we will use later is Stair n :" Dppn ´1, n ´2, ¨¨¨, 1qq.