ALGEBRAIC COMBINATORICS

no. 2
A q-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets
Li, Yifei1
1 University of Illinois at Springfield Department of Mathematical Sciences and Philosophy One University Plaza MS WUIS 13 Springfield Illinois 62703 (USA)
Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 457-469.

Let f(z)=∑ n=0 ∞ (-1) n z n /n!n!. In their 1975 paper, Carlitz, Scoville and Vaughan provided a combinatorial interpretation of the coefficients in the power series 1/f(z)=∑ n=0 ∞ ω n z n /n!n!. They proved that ω n counts the number of pairs of permutations of the nth symmetric group 𝒮 n with no common ascent. This paper gives a combinatorial interpretation of a natural q-analogue of ω n by studying the top homology of the Segre product of the subspace lattice B n (q) with itself. We also derive an equation that is analogous to a well-known symmetric function identity: ∑ i=0 n (-1) i e i h n-i =0, which then generalizes our q-analogue to a symmetric group representation result.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.265
Classification: 05E05, 05E10, 05E18, 05E99, 20C30
Keywords: algebraic combinatorics, poset homology, shellability, symmetric functions, symmetric group representation
Li, Yifei 1

1 University of Illinois at Springfield Department of Mathematical Sciences and Philosophy One University Plaza MS WUIS 13 Springfield Illinois 62703 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2023__6_2_457_0,
     author = {Li, Yifei},
     title = {A $q$-analogue of a result of {Carlitz,} {Scoville} and {Vaughan} via the homology of posets},
     journal = {Algebraic Combinatorics},
     pages = {457--469},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {2},
     year = {2023},
     doi = {10.5802/alco.265},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.265/}
}
TY  - JOUR
AU  - Li, Yifei
TI  - A $q$-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 457
EP  - 469
VL  - 6
IS  - 2
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.265/
DO  - 10.5802/alco.265
LA  - en
ID  - ALCO_2023__6_2_457_0
ER  - 
%0 Journal Article
%A Li, Yifei
%T A $q$-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets
%J Algebraic Combinatorics
%D 2023
%P 457-469
%V 6
%N 2
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.265/
%R 10.5802/alco.265
%G en
%F ALCO_2023__6_2_457_0
Li, Yifei. A $q$-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 457-469. doi : 10.5802/alco.265. https://alco.centre-mersenne.org/articles/10.5802/alco.265/

[1] Björner, Anders Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc., Volume 260 (1980) no. 1, pp. 159-183 | DOI | MR | Zbl

[2] Björner, Anders; Welker, Volkmar Segre and Rees products of posets, with ring-theoretic applications, J. Pure Appl. Algebra, Volume 198 (2005) no. 1-3, pp. 43-55 | DOI | MR | Zbl

[3] Björner, Anders; Wachs, Michelle L. On lexicographically shellable posets, Trans. Amer. Math. Soc., Volume 277 (1983) no. 1, pp. 323-341 | DOI | MR | Zbl

[4] Björner, Anders; Wachs, Michelle L. Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., Volume 348 (1996) no. 4, pp. 1299-1327 | DOI | MR | Zbl

[5] Carlitz, L.; Scoville, Richard; Vaughan, Theresa Enumeration of pairs of permutations, Discrete Math., Volume 14 (1976) no. 3, pp. 215-239 | DOI | MR | Zbl

[6] Gasper, George; Rahman, Mizan Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, Cambridge University Press, Cambridge, 2004, xxvi+428 pages | DOI

[7] Sagan, Bruce E. The symmetric group, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | DOI

[8] Simion, Rodica On q-analogues of partially ordered sets, J. Combin. Theory Ser. A, Volume 72 (1995) no. 1, pp. 135-183 | DOI | MR | Zbl

[9] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages

[10] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI

[11] Sundaram, Sheila Applications of the Hopf trace formula to computing homology representations, Jerusalem combinatorics ’93 (Contemp. Math.), Volume 178, Amer. Math. Soc., Providence, RI, 1994, pp. 277-309 | DOI | MR | Zbl

[12] Sundaram, Sheila The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math., Volume 104 (1994) no. 2, pp. 225-296 | DOI | MR | Zbl

[13] Wachs, Michelle L. Poset topology: tools and applications, Geometric combinatorics (IAS/Park City Math. Ser.), Volume 13, Amer. Math. Soc., Providence, RI, 2007, pp. 497-615 | DOI | MR | Zbl

[14] Wachs, Michelle L. Whitney homology of semipure shellable posets, J. Algebraic Combin., Volume 9 (1999) no. 2, pp. 173-207 | DOI | MR | Zbl

Cited by Sources: