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Eulerian calculus. III: The ubiquitous Cauchy formula. (English) Zbl 0826.05058

Summary: The purpose of this paper is to calculate the distributions of several joint statistics on the symmetric group and related structures that have been studied in the previous two Eulerian calculus papers [ibid. 15, No. 4, 345-362 (1994; Zbl 0811.05069) and ibid. 16, No. 3, 221-252 (1995; Zbl 0822.05066)]. It is shown how the Cauchy formula on symmetric functions, an old MacMahon bijection and a combinatorial lemma on skew- tableaux provide the necessary ingredients for calculating those distributions.

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
05A05 Permutations, words, matrices
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References:

[1] Andrews, George E, The theory of partitions, () · Zbl 0996.11002
[2] George E. Andrews. Private communication, 1994.
[3] Clarke, Robert J; Foata, Dominique, Eulerian calculus, I: univariable statistics, Europ. J. combinatorics, v. 15, 345-362, (1994) · Zbl 0811.05069
[4] Clarke, Robert J; Foata, Dominique, Eulerian calculus, II: an extension of Han’s fundamental transformation, to appear in europ, J. combinatorics, (1994)
[5] Clarke, Robert J; Foata, Dominique, Eulerian calculus, IV: specializations, (1994), in preparation · Zbl 0856.05098
[6] Désarménien, Jacques; Foata, Dominique, Fonctions symétriques et séries hypergéométriques basiques multivariées, Bull. soc. math. France, v. 113, 3-22, (1985) · Zbl 0644.05005
[7] Fine, Nathan J, Basic hypergeometric series and applications, (1988), Amer. Math. Soc Providence · Zbl 0647.05004
[8] Foata, Dominique, LES distributions Euler-mahoniennes sur LES mots, (), preprint, to appear in · Zbl 0829.05058
[9] Gasper, George; Rahman, Mizan, Basic hypergeometric series, () · Zbl 0695.33001
[10] Gessel, Ira, Generating functions and enumeration of sequences, (), 111
[11] Knuth, Donald E, Permutations, matrices, and generalized Young tableaux, Pacific J. math., v. 34, 709-727, (1970) · Zbl 0199.31901
[12] Knuth, Donald E, The art of computer programming, vol. 3, sorting and searching, (1973), Addison-Wesley Reading · Zbl 0302.68010
[13] Macdonald, Ian G, Symmetric functions and Hall polynomials, (1979), Clarendon Press Oxford · Zbl 0487.20007
[14] MacMahon, P.A, The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects, Amer. J. math., v. 35, 314-321, (1913), (Major) · JFM 44.0076.02
[15] MacMahon, P.A, (), (Reprinted by Chelsea, New York, 1955)
[16] MacMahon, P.A, (), (Major)
[17] Rawlings, Don, Multicolored Simon newcomb problems, J. combinatorial theory, v. 53, 53-67, (1990), Ser. A · Zbl 0736.05007
[18] Reiner, Victor, Signed permutation statistics, Europ. J. combinatorics, v. 14, 553-567, (1993) · Zbl 0793.05005
[19] Reiner, Victor, Signed permutations statistics and cycle type, Europ. J. combinatorics, v. 14, 569-579, (1993) · Zbl 0793.05006
[20] Reiner, Victor, Upper binomial posets and signed permutations statistics, Europ. J. combinatorics, v. 14, 581-588, (1993) · Zbl 0793.05007
[21] Stanley, Richard P, Ordered structures and partitions, () · Zbl 0246.05007
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