Multicolored Simon Newcomb problems. (English) Zbl 0736.05007

This paper deals with generating functions for a kind of combinatorial problems known as Simon Newcomb problems. The problems can best be described in terms of shuffling a deck of cards (with certain specifications) and thereafter dividing the deck into piles according to certain rules. The problem is to determine the number of shuffles resulting in \(k+1\) piles. The method used to count these objects consists of mapping the carddeck to a kind of “multicoloured” matrices and then counting these matrices.


05A15 Exact enumeration problems, generating functions
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[1] Andrews, G. E., The theory of compositions II: Simon Newcomb’s problem, Utilitus Math., 7, 33-54 (1975) · Zbl 0326.05010
[2] Andrews, G. E., The Theory of Partitions (1976), Addision-Wesley: Addision-Wesley Reading, MA · Zbl 0371.10001
[3] Carlitz, L.; Roselle, D. P.; Scoville, R. A., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1, 350-374 (1966) · Zbl 0304.05002
[4] Carliz, L., Enumeration of sequences by rises and falls: A refinement of the Simon Newcomb problem, Duke Math. J., 39, 267-280 (1972) · Zbl 0243.05008
[5] Désarménien, J.; Foata, D., Fonctions symétriques et séries hypergeometriques basiques multivariées, Bull. Soc. Math. France, 113, 3-22 (1985) · Zbl 0644.05005
[7] Dillon, J. F.; Roselle, D., Simon Newcomb’s problem, SIAM J. Appl. Math., 17, 1086-1093 (1969) · Zbl 0212.34701
[8] Garsia, A. M.; Gessel, I., Permutation statistics and partitions, Adv. in Math., 31, 288-305 (1979) · Zbl 0431.05007
[9] Gessel, I., Generating Functions and Enumeration of Sequences, MIT doctoral thesis (1977)
[10] Goulden, I. P.; Jackson, D. M., Combinatorial Enumeration (1983), Wiley: Wiley New York · Zbl 0519.05001
[11] MacMahon, P. A., Combinatory Analysis (1915), Cambridge Univ. Press: Cambridge Univ. Press London/New York, (reprinted by Chelsea, New York, 1960) · JFM 45.1271.01
[12] Rawlings, D. P., Generalized Worpitzky identities with applications to permutation enumeration, European J. Combin, 2, 67-78 (1981) · Zbl 0471.05006
[13] Rawlings, D. P., The (q, r)-Simon Newcomb problem, J. Linear Multilinear Algebra, 10 (1981) · Zbl 0516.05004
[14] Riordan, J., An Introduction to Combinatorial Analysis (1959), Wiley: Wiley New York
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