Inclusion-exclusion on Schubert polynomials

We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns 1432 and 1423 is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about principal specializations of Schubert polynomials. We propose a general framework for finding inclusion-exclusion inspired expression of Schubert polynomials of all permutations.


Introduction
Schubert polynomials, introduced by Lascoux and Schützenberger in [15], represent cohomology classes of Schubert cycles in the flag variety.They are also multidegrees of matrix Schubert varieties [12] and wield an impressive collection of combinatorial formulas [1,2,7,9,14,16,19,25].Yet, only recently have their supports been established as integer points of generalized permutahedra [5,20].There has also been several exciting recent developments about the coefficients of Schubert polynomials: (1) they are known to be log-concave along root directions in their Newton polytopes [11]; (2) the set of permutations whose Schubert polynomials have all their coefficients less than or equal to a fixed integer m is closed under pattern containment [6].Recall that π = π 1 . . .π k ∈ S k is a pattern of σ = σ 1 . . .σ n ∈ S n if and only if there are indices 1 ≤ i 1 < i 2 < • • • < i k ≤ n so that the relative order of π 1 , . . ., π k and of σ i1 , . . ., σ i k are the same.where In particular, Theorem 1.1 implies that the set of permutations whose Schubert polynomials have all their coefficients less than or equal to a fixed integer m is closed under pattern containment.
The first result of this paper is a broad extension of Theorem 1.1 for 1432 and 1423 avoiding permutations: Theorem 1.2.Let w ∈ S n be a 1432 and 1423 avoiding permutation and let u be a subword of w.Then where M w,v := (i,j)∈D(w)\ D(w) v x i .
In Theorem 1.2 we use the relation of containment on words: for words u, v, we say u ≤ v if u occurs as a subword in v.Moreover, for a word v of length n, π = perm(v) is the permutation in S n such that the relative order of π 1 , . . ., π n and of v 1 , . . ., v n are the same.For these and other definitions used in Theorems 1.1 and 1.2 see Sections 2 and 3 which lay them out in detail.Here we give an example of Theorem 1.2 for illustration.For w = 1342 and u = 42 we have {v | u ≤ v ≤ w} = {1342, 142, 342, 42}, so the alternating sum in (2) becomes M w,1342 S 1342 (x 1 , x 2 , x 3 , x 4 ) − M w,142 S 132 (x 1 , x 3 , x 4 ) − M w,342 S 231 (x 2 , x 3 , x 4 ) + M w,42 S 21 (x 3 , x 4 ) which indeed has nonnegative coefficients.See Figure 3 for an illustration.
An immediate corollary of Theorem 1.2 is the following theorem: Theorem 1.3.Let w ∈ S n be a 1432 and 1423 avoiding permutation.If u is a subword of w, then where S perm(v) (1) denotes the value of the Schubert polynomial S perm(v) with all its variables set to 1.
Theorem 1.3 is closely related to a recent conjecture of Gao [10,Conjecture 3.2] regarding the principal specialization of Schubert polynomials as we now explain.We also conjecture (Conjecture 5.1) in Section 5 that Theorem 1.3 holds for all permutations w ∈ S n .
1.2.Principal specializations of Schubert polynomials.Macdonald [17,Eq. 6.11] famously expressed the principal specialization S σ (1) of the Schubert polynomial S σ in terms of the reduced words of σ.Fomin and Kirillov [8] placed this expression in the context of plane partitions for dominant permutations, while after two decades Billey et al. [3] provided a combinatorial proof.In 2017, Stanley [23] considered the asymptotics of S σ (1) as well as the role pattern containment plays in its value.The asymptotics question was partially answered by Morales, Pak and Panova [21], while the pattern avoidance question inspired Weigandt [24] and Gao [10], among others, to seek an understanding of S σ (1) in terms of the permutation patterns of σ.Weigandt showed that S σ (1) ≥ 1 + p 132 (σ), where p π (σ) is the number of patterns π in the permutation σ, while Gao improved this to S σ (1) ≥ 1 + p 132 (σ) + p 1432 (σ).Gao conjectured that there exist nonnegative integers c w , for w ∈ S ∞ , such that where v ≤ w denotes that v occurs as a subword in w.

It follows readily via inclusion-exclusion that for
Thus, Theorem 1.3 settles Gao's conjecture 1.4 for 1432 and 1423 avoiding permutations w ∈ S ∞ when we specialize it to the empty word u = ().Moreover, we also provide a combinatorial interpretation of the numbers c w for 1432 and 1423 avoiding permutations w ∈ S ∞ : Theorem 1.5.For 1432 and 1423 avoiding permutations w ∈ S ∞ the value of c w is the number of diagrams C ≤ D(w) that cannot be written as C aug for some C ≤ D(w).
See Section 3.3 for more details.
See Section 4 for the definition of the set of diagrams P k,σ k (D(σ)) used in the statement of Theorem 1.6 above and Section 5 for a discussion of how Theorem 1.6 could be used to generalize Theorem 1.2 as well as Conjecture 5.4 examining the strength of Theorem 1.6.
Outline of this paper.Section 2 lays out the general background on Schubert polynomials that we rely on.Section 3 contains the setup and proofs of Theorem 1.2, 1.3 and 1.5.Section 4 provides a proof of Theorem 1.6 and its generalization Theorem 4.1, while Section 5 concludes with conjectures and open problems.

Background on Schubert polynomials
Schubert polynomials were originally defined via divided difference operators.We will instead define them as dual chatacters of flagged Weyl modules for Rothe diagrams.This section follows the exposition of [5,6].We interchangeably think of D as a collection of boxes (i, j) in a grid, viewing an element i ∈ C j as a box in row i and column j of the grid.When we draw diagrams, we read the indices as in a matrix: i increases top-to-bottom and j increases left-to-right.
The Rothe diagram D(w) of a permutation w ∈ S n is the diagram ) j and j < w i }.
Note that Rothe diagrams have the northwest property: If (r, c ), (r , c) ∈ D(w) with r < r and c < c , then (r, c) ∈ D(w).
Let G = GL(n, C) be the group of n × n invertible matrices over C and B be the subgroup of G consisting of the n × n upper-triangular matrices.The flagged Weyl module is a representation M D of B associated to a diagram D. The dual character of M D has been shown in certain cases to be a Schubert polynomial [13] or a key polynomial [22].We will use the construction of M D in terms of determinants given in [18].
Denote by Y the n × n matrix with indeterminates y ij in the upper-triangular positions i ≤ j and zeros elsewhere.Let C[Y ] be the polynomial ring in the indeterminates For any B-module N , the character of N is defined by char(N )(x 1 , . . ., x n ) = tr (X : N → N ), where X is the diagonal matrix diag(x 1 , x 2 , . . ., x n ) with diagonal entries x 1 , . . ., x n , and X is viewed as a linear map from N to N via the B-action.Define the dual character of N to be the character of the dual module N * :

Results about dual characters of flagged Weyl modules.
A special case of dual characters of flagged Weyl modules of diagrams are Schubert polynomials: 13]).For w a permutation and D(w) its Rothe diagram we have that the Schubert polynomial Let C (1) , . . ., C (r) be all the diagrams C such that C ≤ D and In light of the last inequality, it is natural to wonder when equality holds.This is what Fan & Guo [4] did: Then, for a permutation w ∈ S n , if and only if w avoids the patterns 1432 and 1423.
In particular, Theorem 2.6 implies: 3. Proof of Theorems 1.2, 1.3 and 1.5 In this section we prove Theorems 1.2, 1.3 and 1.5.We start by giving the necessary definitions and lemmas.
is the permutation in S n such that the relative order of π 1 , . . ., π n and of v 1 , . . ., v n are the same.
Thus perm(v) = (51423) −1 = 24531.Notice that we can obtain perm(v) from v by replacing the smallest character of v with 1, the second smallest with 2, and so on.Definition 3.5.Let w ∈ S n and let v be a subword of w.We define Example 3.6.Let w = 134265 and v = 3265.Then Notice that the resulting indices will always be in ascending order.See Figure 1 for an illustration.
Definition 3.8.Suppose C ≤ D(w) for some permutation w ∈ S n .Then, for any subword v ≤ w, we define C v to be the diagram obtained by keeping only the boxes in the rows corresponding to v.That is, 3.2.Theorem 1.2 and its proof.
Theorem 1.2.Let w ∈ S n be a 1432 and 1423 avoiding permutation and let u be a subword of w.Then (5) where M w,v := which indeed has nonnegative coefficients.See Figure 2 for an illustration.
which indeed has nonnegative coefficients.See Figure 3 for an illustration.
To aid the proof of Theorem 1.2 we extend Corollary 2.7 to words: Lemma 3.11.Let w ∈ S n be a 1432 and 1423 avoiding permutation, and let v be a subword of w. .These in turn yield the Schubert polynomials in the expression (7).The red numbers on the left of the diagrams signify these subwords; the blue numbers are the row numbers yielding the variables of the corresponding Schubert polynomials in the expression (7).The purple boxes correspond to the boxes of the Rothe diagram of w = 1342 that are removed in order to obtain v; graphically these are the boxes struck by yellow if the yellow highlighted rows and columns are extended; the row indices of these boxes yield the monomials M w,v .
Proof.Fix m, and let x i = m and C's boxes all lie in rows K If m is divisible by some x i where i / ∈ K, then the coefficient of m in S perm(v) (x w −1 (v) ) is 0, and no diagram C with boxes only in rows K can ever satisfy (i,j)∈C x i = m, so |A| = 0 and we are done.
Let which is simply m under the reindexing x ki → x i .Since w is 1432 and 1423 avoiding, so are v and perm(v), thus by Corollary 2.7, the coefficient of m in S perm(v) (x 1 , . . ., x |v| ) is equal to |B|, where Consider the function f : B → A given by Notice that the boxes of f (C) all lie in rows K, and since (i,j)∈C x i = m , we have (i,j)∈f (C) x i = m.Furthermore, from the definition of Rothe diagrams, if (i, j) f is clearly injective by construction.To see that it is surjective, note that if C ∈ A, then C ≤ D(w) v , so every box in C is of the form (k i , v(j)) and the diagram Proof of Theorem 1.2.We must show that for every monomial m, Consider the two families of sets Since v ≤ w, K v ⊆ K w , and also the boxes of D(w) \ D(w) v all lie in rows K w \ K v .Thus, D(w) \ D(w) v is disjoint from every C ∈ A v , and so there is an obvious injection f from A v to B v defined by and so it suffices to show that ( 10) This quantity is necessarily non-negative, as desired.
By setting all x i 's to 1 in Theorem 1.2 we obtain: We conjecture (Conjecture 5.1) that Theorem 1.3 generalizes to all permutations w ∈ S n .
3.3.Gao's conjecture 1.4, Theorem 1.5 and its proof.Gao [10] defined a sequence of integers {c u } m≥1,u∈Sm recursively, as follows: where |u| = m if u ∈ S m , and p u (w) is the number of occurrences of u as a pattern in w.
Gao showed that c w = 0 whenever w(n) = n, so the definition of c w can be extended to all w ∈ S ∞ .In the same paper, he conjectured the following: Conjecture 3.12.([10, Conjecture 3.2]) We have c w ≥ 0 for all w ∈ S ∞ .
Notice that p u (w) = #{words v such that u = perm(v) and v ≤ w}.Thus, we can rewrite ( 14) as (15) c w = S w (1) where the −1 has been absorbed into the sum as c () .Note that this perspective explains the equivalence of Conjectures 1.4 and 3.12.By inclusion-exclusion, ( 15) is equivalent to By tracing the proof of Theorem 1.2 for the case u = (), we can obtain an interpretation of the coefficient of m in v≤w (−1) |w|−|v| M w,v S v (x w −1 (v) ) in terms of augmentations of diagrams C ≤ D(w).In particular, we readily obtain: Theorem 1.5.For 1432 and 1423 avoiding permutations w ∈ S ∞ the value of c w is the number of diagrams C ≤ D(w) that cannot be written as C aug for some C ≤ D(w).
We conclude this section by illustrating Theorem 1.5 for permutations 1342 and 12453.The main result of this section is a generalization of Theorems 1.1 and 1.6:  • there is some D) to be the smallest set satisfying the following: D), and In this case, these are all the monomials M for which See Figure 4 for an illustration.
so in this case the monomials prescribed by Theorem 1.6 are not the only monomials that could work.See Figure 5 for an illustration.be a set of diagrams with C (i) ≤ D for each i, and denote are linearly dependent, then so are the polynomials .
Proof.We are given that ( 17) (17) can be rewritten as (18) det However, since det(Y K l D l ) = 0, we conclude that ( 19) First consider the case that the only boxes of D in row k or column l are those in D l .If this is the case then ( 20) Combining (19) and (21) we obtain that the polynomials j∈[n]\{l} det(Y are dependent, as desired.Now, suppose that there are boxes of D in row k that are not in D l .Let j 1 < • • • < j p be all indices j = l such that D j = D j ∪ {k}.Then, for each i ∈ [m] and q ∈ [p], C (i) jq \ C (i) jq = K jq .For each q ∈ [p], let k q be the only element of K jq ; then (19) implies that as is seen by Laplace expansion on the k q th row of det(Y C (i) jq Dj q ), and therefore Thus, (22) and (24) imply that (25) We are now ready to prove Theorem 4.1: Proof of Theorem 4.1 Let M = M (x 1 , . . ., x n ).Suppose there is some K Note the inequality, which is because we have only a subset of the C.
Finally, Lemma 4.7 implies that dim

Problems and Conjectures
The results of this paper naturally give rise to the following Conjectures and Problems.

2. 1 .
Definition of dual characters of flagged Weyl modules.A diagram is a sequence D = (C 1 , C 2 , . . ., C n ) of finite subsets of [n], called the columns of D.
the submatrix of Y obtained by restricting to rows R and columns S. For R, S ⊆ [n], we say R ≤ S if #R = #S and the kth least element of R does not exceed the kth least element of S for each k.For any diagrams C = (C 1 , . . ., C n ) and D = (D 1 , . . ., D n ), we say C ≤ D if C j ≤ D j for all j ∈ [n].Definition 2.1.For a diagram D = (D 1 , . . ., D n ), the flagged Weyl module M D is defined by

Figure 1 .
Figure 1.The left diagram is the Rothe diagram of the permutation w = 134265 (the permutation w is noted in red to the left of the diagram).The row indices are noted in blue to the left of the diagram.The right diagram shows the subword v = 3265 of w = 134265 graphically: it is obtained by removing the yellow highlighted rows and columns from the Rothe diagram of w.The indices w −1 (v) shown in blue to the left of the diagram are simply the row indices corresponding to this graphical presentation of the subword v = 3265 of w = 134265.Definition 3.7.Given a diagram D ⊆ [n] × [n] and sets of indices K, L ⊆ [n] with #K = #L, let D K,L denote the diagram obtained from D by keeping only the boxes in rows K and columns L:

Figure 2 .
Figure 2. The four diagrams in this figure correspond left to right to the subwords {v | u ≤ v ≤ w} = {2143, 143, 243, 43} for w = 2143 and u = 43 as in Example 3.9.These in turn yield the Schubert polynomials in the expression(6).The red numbers on the left of the diagrams signify these subwords; the blue numbers are the row numbers yielding the variables of the corresponding Schubert polynomials in the expression(6).The purple boxes correspond to the boxes of the Rothe diagram of w = 2143 that are removed in order to obtain v; graphically these are the boxes struck by yellow if the yellow highlighted rows and columns are extended; the row indices of these boxes yield the monomials M w,v .

x 2 •Figure 3 .
Figure 3.The four diagrams in this figure correspond left to right to the subwords {v | u ≤ v ≤ w} = {1342, 142, 342, 42} for w = 1342 and u = 42 as in Example 3.10.These in turn yield the Schubert polynomials in the expression(7).The red numbers on the left of the diagrams signify these subwords; the blue numbers are the row numbers yielding the variables of the corresponding Schubert polynomials in the expression(7).The purple boxes correspond to the boxes of the Rothe diagram of w = 1342 that are removed in order to obtain v; graphically these are the boxes struck by yellow if the yellow highlighted rows and columns are extended; the row indices of these boxes yield the monomials M w,v .
the elements of K.By the previous discussion, we may as well assume that we and for all u ≤ v, v ≤ w, B u ∩ B v = B u∧v , where u ∧ v denotes the maximal word contained in both u and v. Let I = {v | u ≤ v ≤ w and |v| = |w| − 1}.Then, using inclusion-exclusion, we find that u≤v≤w

(− 1 )Theorem 3 . 13 .Definition 3 . 14 .
|w|−|v| S v (1).Thus, Theorem 1.3 immediately implies: Conjecture 3.12 (equivalently, Conjecture 1.4) holds for 1432 and 1423 avoiding permutations w ∈ S ∞ .Moreover, Theorem 1.5 below provides a combinatorial interpretation for c w when w is 1432 and 1423 avoiding.Given diagrams C, D ⊆ [n] × [n] and k, l ∈ [n], let C and D denote the diagrams obtained from C and D by removing any boxes in row k or column l.Fix a diagram D. For each diagram C, let its augmentation with respect to the diagram D be:

Theorem 4 . 1 .
Fix a diagram D ⊆ [n] × [n] and let D be the diagram obtained from D by removing any boxes in row k or column l.If there is some diagram K

Theorem 1 .Definition 4 . 2 .
6 is a special case of Theorem 4.1 when D is a Rothe diagram of a permutation.We now proceed to define the set of diagrams P k,l (D) used in the statement of Theorem 4.1 above.Fix a diagram D ⊆ [n] × [n] and integers k, l ∈ [n].Define Purple k,l (D) to be the set of boxes (i, j) such that:

Problem 5 . 5 .
Let w ∈ S n and let u be a subword of w.Using the monomials from Theorem 1.6 (or its extension asked for in Problem 5.3) is it possible to pick suitable monomials m w,v ∈ Z[x 1 , . . ., x n ] to make the expressionu≤v≤w (−1) |w|−|v| m w,v S perm(v) (x w −1 (v) ) belong to Z ≥0 [x 1 , . . ., x n ]?Note that a positive answer to Problem 5.5 would be an extension of Theorem 1.2 which would readily imply Conjecure 5.1 as well as Gao's Conjecture 1.4.
1.3.Extending Theorems 1.1 & 1.2.Both Theorem 1.3 and Theorem 1.5 are byproducts of our main Theorem 1.2.It is thus most natural to ask in what generality Theorem 1.2 holds.While Theorem 1.3 is conjectured by Gao to hold for all permutations, Theorems 1.2 and 1.5 as stated do not.Theorem 1.2 fails already for w = 1432.However, the reason it fails leads to other possibilities: the monomials M w,v we used to formulate Theorem 1.2 are inspired by Theorem 1.1 and are one of many choices we might have made.Fix σ ∈ S n and let π ∈ S n−1 be the pattern of σ with Rothe diagram D(π) obtained by removing row k and column σ k from D(σ).If there is some diagram