×

Eulerian calculus. II: An extension of Han’s fundamental transformation. (English) Zbl 0822.05066

Summary: In our first Eulerian calculus paper [ibid. 15, No. 4, 345-362 (1994; Zbl 0811.05069)], we introduced the \(k\)-descent and \(k\)-exceedance statistics on words, considered various bijections associated with them and calculated their distributions. We consider the \(k\)-major and \(k\)-den statistics and show that the joint des-maj and exc-den distributions of that \(k\)-extension are identical. We also consider the generating functions for these statistics.

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05A15 Exact enumeration problems, generating functions
68R15 Combinatorics on words
20B30 Symmetric groups

Citations:

Zbl 0811.05069
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Andrews, George E, On the foundations of combinatorial theory V, (), 345-375 · Zbl 0237.05002
[2] Carlitz, L, q-Bernoulli and Eulerian numbers, Trans. amer. math. soc., v. 76, 332-350, (1954) · Zbl 0058.01204
[3] Carlitz, L, Eulerian numbers and polynomials, Math. magazine, v. 33, 247-260, (1959) · Zbl 0092.06601
[4] Carlitz, L, A combinatorial property of q-Eulerian numbers, Amer. math. monthly, v. 82, 51-54, (1975) · Zbl 0296.05007
[5] Cartier, Pierre; Foata, Dominique, Problèmes combinatoires de permutations et réarrangements, () · Zbl 1364.05001
[6] Clarke, Robert J; Foata, Dominique, Eulerian calculus, I: univariable statistics, Europ. J. combinatorics, v. 15, 345-362, (1994) · Zbl 0811.05069
[7] Clarke, Robert J; Foata, Dominique, Eulerian calculus, III: the ubiquitous Cauchy formula, Europ. J. combinatorics, (1994), to appear in · Zbl 0826.05058
[8] Denert, Marlene, The genus zeta function of hereditary orders in central simple algebras over global fields, Math. comp., v. 54, 449-465, (1990) · Zbl 0687.16003
[9] Foata, Dominique, LES distributions Euler-mahoniennes sur LES mots, (), published by · Zbl 0829.05058
[10] Foata, Dominique; Zeilberger, Doron, Denert’s permutation statistic is indeed Euler-Mahonian, Studies in appl. math., v. 83, 31-59, (1990) · Zbl 0738.05001
[11] Garsia, Adriano M, On the “maj“ and “inv” q-analogues of Eulerian polynomials, Linear and multilinear algebra, v. 8, 21-34, (1979) · Zbl 0442.05002
[12] Garsia, Adriano M; Gessel, Ira, Permutation statistics and partitions, Advances in math., v. 31, 288-305, (1979) · Zbl 0431.05007
[13] Gessel, Ira, Generating functions and enumeration of sequences, (), 111
[14] Han, Guo-Niu, Une transformation fondamentale sur LES réarrangements de mots, Advances in math., v. 105, 26-41, (1994) · Zbl 0798.05001
[15] Lothaire, M, Combinatorics on words, () · Zbl 1001.68093
[16] MacMahon, P.A, (), (Reprinted by Chelsea, New York, 1955)
[17] Rawlings, Don, Multicolored Simon newcomb problems, J. combinatorial theory, ser. A, v. 53, 53-67, (1990) · Zbl 0736.05007
[18] Schützenberger, Marcel-Paul, On a factorization of free monoids, (), 21-24 · Zbl 0219.20039
[19] Stanley, Richard P, Ordered structures and partitions, () · Zbl 0246.05007
[20] Stanley, Richard P, Binomial posets, (), 336-356 · Zbl 0331.05004
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.