Some identities involving \(q\)-Stirling numbers of the second kind in type B. (English) Zbl 1533.05027
Summary: The recent interest in type B \(q\)-Stirling numbers of the second kind prompted us to give a type B analogue of a classical identity connecting the \(q\)-Stirling numbers of the second kind and Carlitz’s major \(q\)-Eulerian numbers, which turns out to be a \(q\)-analogue of an identity due to E. Bagno et al. [ibid. 26, No. 3, Research Paper P3.9, 20 p. (2019; Zbl 1416.05037)]. We provide a combinatorial proof of this identity and an algebraic proof of a more general identity for colored permutations. In addition, we prove some \(q\)-identities about the \(q\)-Stirling numbers of the second kind in types A, B and D.
MSC:
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05A05 | Permutations, words, matrices |
| 05A30 | \(q\)-calculus and related topics |
| 11B73 | Bell and Stirling numbers |
Citations:
Zbl 1416.05037References:
| [1] | R. M. Adin, Y. Roichman, The flag major index and group actions on polynomial rings, European J. Combin. 22 (4) (2001) 431-446. · Zbl 1058.20031 |
| [2] | G. E. Andrews, The theory of partitions, Reprint of the 1976 original, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. |
| [3] | E. Bagno, R. Biagioli, D. Garber, Some identities involving second kind Stirling numbers of types B and D, Electron. J. Combin. 26(3):#P3.9, (2019). · Zbl 1439.05031 |
| [4] | E. Bagno, D. Garber, T. Komatsu, A q, r-analogue for the Stirling numbers of the second kind of Coxeter groups of type B, Pure Math. Appl. (PU.M.A.) 30 (1) (2022) 8-16. · Zbl 1513.05040 |
| [5] | M. Bóna, Combinatorics of Permutations, Chapman & Hall\CRC, 2004. · Zbl 1052.05001 |
| [6] | R. Biagioli, J. Zeng, Enumerating wreath products via Garsia-Gessel bijections, Eu-ropean J. Combin. 32 (4) (2011) 538-553. · Zbl 1229.05013 |
| [7] | L. Carlitz, q-Bernouli numbers and polynomials, Duke Math. J. 15 (1948) 987-1000. · Zbl 0032.00304 |
| [8] | C.-O. Chow, I. M. Gessel, On the descent numbers and major indices for the hyper-octahedral group, Adv. in Appl. Math. 38 (2007) 275-301. · Zbl 1124.05005 |
| [9] | C.-O. Chow, T. Mansour, A Carlitz identity for the wreath product Z r ≀ S n , Adv. in Appl. Math. 47 (2011) 199-215. · Zbl 1234.05020 |
| [10] | I. Dolgachev, V. Lunts, A character formula for the representation of the Weyl group in the cohomology of the associated toric variety, J. Algebra 168 (3) (1994) 741-772. · Zbl 0813.14040 |
| [11] | D. Foata, Eulerian polynomials: from Euler’s time to the present. The legacy of Alladi Ramakrishnan in the mathematical sciences, 253-273, Springer, New York, 2010. · Zbl 1322.01007 |
| [12] | A. M. Garsia, On the “maj” and “inv” q-analogue of Eulerian numbers, Linear Mul-tilinear Algebra 8 (1979) 21-34. · Zbl 0442.05002 |
| [13] | M. Ishikawa, A. Kasraoui, J. Zeng, Euler-Mahonian statistics on ordered set parti-tions, SIAM J. Discrete Math. 22 (3) (2008) 1105-1137. · Zbl 1189.05021 |
| [14] | A. Kasraoui, J. Zeng, Euler-Mahonian statistics on ordered set partitions. II, J. Combin. Theory Ser. A 116 (3) (2009) 539-563. · Zbl 1228.05054 |
| [15] | T. Komatsu, E. Bagno, D. Garber, Analytic aspects of q, r-analogue of poly-Stirling numbers of both kinds, arXiv:2209.06674v2. |
| [16] | G. Ksavrelof, J. Zeng, Nouvelles statistiques de partitions pour les q-nombres de Stirling de seconde espèce, Discrete Math. 256 (3) (2002) 743-758. · Zbl 1007.05016 |
| [17] | P. A. MacMahon, Combinatory Analysis, Two volumes (bound as one) Chelsea Pub-lishing, Co., New York, 1960. · Zbl 0101.25102 |
| [18] | T. K. Petersen, Eulerian Numbers, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser/Springer, New York, 2015. · Zbl 1337.05001 |
| [19] | J. B. Remmel, M. Wachs, Rook Theory, Generalized Stirling numbers and (p, q)-analogues, Electron. J. Combin. 11(1):#R84 (2004). · Zbl 1065.05018 |
| [20] | J. B. Remmel, A. T. Wilson, An extension of MacMahon’s equidistribution theorem to ordered set parptitions, J. Combin. Theory Ser. A 134 (2015) 242-277. · Zbl 1315.05019 |
| [21] | B. E. Sagan, J. P. Swanson, q-Stirling numbers in type B, European J. Combin. 118 (2024), Paper No. 103899. · Zbl 1535.05038 |
| [22] | R. Stanley, Enumerative Combinatorics, vol. I, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. · Zbl 0889.05001 |
| [23] | E. Steingrímsson, Statistics on ordered partitions of sets, J. Comb. 11 (3) (2020) 557-574. · Zbl 1441.05023 |
| [24] | R. Suter, Two analogues of a classical sequence, J. Int. Seq. 3 (2000), Article 0.1.8, 13pp. · Zbl 1004.05002 |
| [25] | J. P. Swanson, N. R. Wallach, Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds, Comb. Theory 3 (3) (2023), #17. · Zbl 1534.20058 |
| [26] | T. Zaslavsky, The Geometry of Root Systems and Signed Graphs, Amer. Math. Monthly 88 (2) (1981) 88-105. · Zbl 0466.05058 |
| [27] | J. Zeng, C. G. Zhang, A q-analog of Newton’s series, Stirling functions and Eulerian functions, Results Math. 25 (3-4) (1994) 370-391. · Zbl 0816.33010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
