Stanley, Richard P. Binomial posets, Möbius inversion, and permutation enumeration. (English) Zbl 0331.05004 J. Comb. Theory, Ser. A 20, 336-356 (1976). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 3 ReviewsCited in 73 Documents MSC: 05A15 Exact enumeration problems, generating functions 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 05A19 Combinatorial identities, bijective combinatorics 11A25 Arithmetic functions; related numbers; inversion formulas 06A06 Partial orders, general PDFBibTeX XMLCite \textit{R. P. Stanley}, J. Comb. Theory, Ser. A 20, 336--356 (1976; Zbl 0331.05004) Full Text: DOI Online Encyclopedia of Integer Sequences: Triangle read by rows: T(n,k) is the number of even permutations of {1,2,...,n} having k descents (n >= 1, k >= 0). Triangle read by rows: T(n,k) is the number of odd permutations of {1,2,...,n} having k descents. (n>=1, k>=1). The number of pairs of permutations in the product group S_n X S_n with k common descents, n >= 1 and 0 <= k <= n-1. References: [1] André, D., Developments de sec \(x\) et de tang \(x\), C. R. Acad. Sci. Paris, 88, 965-967 (1879) · JFM 11.0187.01 [2] deBruijn, N. G., Permutations with given ups and downs, Nieuw Arch. Wisk, 18, 3, 61-65 (1970) · Zbl 0203.30601 [3] Carlitz, L., \(q\)-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., 76, 332-350 (1954) · Zbl 0058.01204 [4] Carlitz, L., Permutations with prescribed pattern, Math. Nachr., 58, 31-53 (1973) · Zbl 0229.05015 [5] Carlitz, L., Enumeration of up-down sequences, Discrete Math., 4, 273-286 (1973) · Zbl 0255.05003 [6] Carlitz, L., Discrete Math., 5, 291 (1973), Addendum [7] Carlitz, L., Permutations and sequences, Advances in Math, 14, 92-120 (1974) · Zbl 0285.05011 [8] Carlitz, L.; Scoville, R.; Vaughan, T., Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc., 80, 881-884 (1974) · Zbl 0291.05007 [9] Comtet, L., Advanced Combinatorics (1974), Reidel: Reidel Dordrecht/Boston [10] Dillon, J. F.; Roselle, D. P., Simon Newcomb’s problem, SIAM J. Appl. Math., 17, 1086-1093 (1969) · Zbl 0212.34701 [11] Doubilet, P.; Rota, G.-C; Stanley, R. P., On the foundations of combinatorial theory (VI): The idea of generating function, (Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory (1972), University of California), 267-318 [12] Foata, D.; Schützenberger, M. P., Théorie Géométrique des Polynômes Eulériens, (Lecture Notes in Mathematics, 138 (1970), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0214.26202 [13] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1960), Oxford · Zbl 0086.25803 [14] MacMahon, P. A., Combinatory Analysis, Vols. 1-2 (1916), reprinted by Chelsea, New York, 1960 [15] Niven, I., A combinatorial problem of finite sequences, Nieuw Arch. Wisk., 16, 3, 116-123 (1968) · Zbl 0164.33102 [16] Riordan, J., An Introduction to Combinatorial Analysis (1958), Wiley: Wiley New York · Zbl 0078.00805 [17] Rota, G.-C, On the foundations of combinatorial theory, I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie, 2, 340-368 (1964) · Zbl 0121.02406 [18] Stanley, R. P., Ordered structures and partitions, Mem. Amer. Math. Soc., 119 (1972) · Zbl 0246.05007 [19] Stanley, R. P., Supersolvable lattices, Algebra Universalis, 2, 197-217 (1972) · Zbl 0256.06002 [20] R. P. Stanley; R. P. Stanley 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.