Tanimoto, Shinji Parity alternating permutations and signed Eulerian numbers. (English) Zbl 1233.05217 Ann. Comb. 14, No. 3, 355-366 (2010). Summary: This paper introduces subgroups of the symmetric group and studies their combinatorial properties. Their elements are called parity alternating, because they are permutations with even and odd entries alternately. The objective of this paper is twofold. The first is to derive several properties of such permutations by subdividing them into even and odd permutations. The second is to discuss their combinatorial properties; among others, relationships between those permutations and signed Eulerian numbers. Divisibility properties by prime powers are also deduced for signed Eulerian numbers and several related numbers. Cited in 7 Documents MSC: 05E15 Combinatorial aspects of groups and algebras (MSC2010) 20B35 Subgroups of symmetric groups Keywords:Eulerian numbers; recurrence; permutations PDFBibTeX XMLCite \textit{S. Tanimoto}, Ann. Comb. 14, No. 3, 355--366 (2010; Zbl 1233.05217) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Triangle a(n,k) (1<=k<=n) of ”signed Eulerian numbers” read by rows. a(n) = 2(m!)^2 for n = 2m and m!(m+1)! for n = 2m+1. Tanimoto triangle read by rows: T(n,k) = number of ”parity-alternating permutations” (PAPS) of n symbols with k ascents. Number of parity preserving permutations of the set {1,2,...,n} with exactly k cycles. References: [1] Désarménien J., Foata D.: The signed Eulerian numbers. Discrete Math. 99(1-3), 49–58 (1992) · Zbl 0769.05094 [2] Foata D., Schützenberger M.-P.: Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Mathematics, Vol. 138. Springer-Verlag, Berlin (1970) [3] Graham R.L., Knuth D.E., Patashnik O.: Concrete Mathematics. Addison-Wesley Publishing Co., Reading, MA (1989) · Zbl 0668.00003 [4] Kerber A.: Algebraic Combinatorics via Finite Group Actions. Bibliographisches Institut, Mannheim (1991) · Zbl 0726.05002 [5] Knuth D.E.: The Art of Computer Programming, Vol. 3. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills (1973) · Zbl 0191.17903 [6] Lesieur L., Nicolas J.-L.: On the Eulerian numbers M n = max1 A(n,k). European J. Combin. 13(5), 379–399 (1992) · Zbl 0766.11012 [7] Loday J.-L.: Opérations sur l’homologie cyclique des algèbres commutatives. Invent. Math. 96(1), 205–230 (1989) · Zbl 0686.18006 [8] Mantaci R.: Binomial coefficients and anti-exceedances of even permutations: a combinatorial proof. J. Combin. Theory Ser. A 63(2), 330–337 (1993) · Zbl 0777.05002 [9] Regev A., Roichman Y.: Permutation statistics on the alternating group. Adv. Appl.Math. 33(4), 676–709 (2004) · Zbl 1057.05004 [10] Tanimoto S.: An operator on permutations and its application to Eulerian numbers. European J. Combin. 22(4), 569–576 (2001) · Zbl 0986.05005 [11] Tanimoto S.: A study of Eulerian numbers by means of an operator on permutations. European J. Combin. 24(1), 33–43 (2003) · Zbl 1014.05007 [12] Tanimoto, S.: A study of Eulerian numbers for permutations in the alternating group. Integers 6, #A31 (2006) · Zbl 1104.11011 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.