## 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

Zbl 0811.05069
Full Text:

### 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
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