We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the former is a generalization of a quasisymmetric function introduced by Grinberg. We prove representation-theoretic analogues of classical and recent results, including F-positivity, and combinatorial reciprocity theorems. We deduce results for orbital quasisymmetric functions, and study a generalization of the notion of strongly flawless sequences.
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Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series, vol. 29, American Mathematical Society, Providence, RI, 2010, With forewords by Kenneth Brown and Stephen Chase and André Joyal. MR2724388 (2012g:18009)
Per Alexandersson and Robin Sulzgruber, P-partitions and p-positivity, Sém. Lothar. Combin. 82B (2020), Art. 61, 12. MR4098282
Cristina Ballantine, Zajj Daugherty, Angela Hicks, Sarah Mason, and Elizabeth Niese, On quasisymmetric power sums, J. Combin. Theory Ser. A 175 (2020), 105273
Matthias Beck and Raman Sanyal, Combinatorial reciprocity theorems, Graduate Studies in Mathematics, vol. 195, American Mathematical Society, Providence, RI, 2018, An invitation to enumerative geometric combinatorics. MR3839322
Peter J. Cameron, K. K. Kayibi, Orbital chromatic and flow roots, Combin. Probab. Comput. 16 (2007), 401–407.
N. G. de Bruijn, Ca. van Ebbenhorst Tengbergen, D. Kruyswijk, On the set of divisors of a number, Nieuw Arch. Wiskunde (2) 23 (1951), 191–193.
Brittney Ellzey, Chromatic quasisymmetric functions of directed graphs, Sém. Lothar. Combin. 78B (2017), Art. 74, 12. MR3678656
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991, A first course, Readings in Mathematics. MR1153249
Ira M. Gessel, Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289–317. MR777705
Darij Grinberg, Double posets and the antipode of QSym, Electron. J. Combin. 24 (2017), no. 2, Paper 2.22, 47. MR3650271
Takayuki Hibi, What can be said about pure O-sequences?, J. Combin. Theory Ser. A 50 (1989), no. 2, 319–322.
June Huh, h-vectors of matroids and logarithmic concavity, Adv. Math. 270 (2015), 49–59.
Katharina Jochemko, Order polynomials and Pólya’s enumeration theorem, Electron. J. Combin. 21 (2014), no. 2, Paper 2.52, 11. MR3244818
Martina Juhnke-Kubitzke and Dinh Van Le, Flawlessness of h -vectors of broken circuit complexes, Int. Math. Res. Not. IMRN (2018), no. 5, 1347–1367. MR3801465
Claudia Malvenuto and Christophe Reutenauer, A self paired Hopf algebra on double posets and a Littlewood-Richardson rule, J. Combin. Theory Ser. A 118 (2011), no. 4, 1322–1333.
John Shareshian and Michelle L. Wachs, Chromatic quasisymmetric functions, Adv. Math. 295 (2016), 497–551. MR3488041
Richard P. Stanley, Combinatorial reciprocity theorems, Adv. Math. 14 (1974), 194–253. MR411982
Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. 111 (1995), no. 1, 166–194. MR1317387 (96b:05174)
Alan Stapledon, Equivariant Ehrhart theory, Adv. Math. 226 (2011), no. 4, 3622–3654. MR2764900
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Communicated by Vasu Tewari.
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White, J.A. The Chromatic Quasisymmetric Class Function of a Digraph. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00556-1
- Chromatic polynomials
- Quasisymmetric functions
- Poset partitions
- Group actions
Mathematics Subject Classification