This article gives a brief introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other

The definitions of descent, excedance, major index, inversion index and Denert's statistic for the elements of the symmetric group S d are generalized to indexed permutations, i.e. the elements of the group S n d := Z n o S d , where o is wreath product with respect to the usual action of S d by permutations of f1; 2; : : : ; dg. It is shown, bijectively, that excedances and descents are equidistributed, and the corresponding descent polynomial, analogous to the Eulerian polynomial, is computed as the f--eulerian polynomial of a simple polynomial. The descent polynomial is shown to equal the h--polynomial (essentially the h--vector) of a certain triangulation of the unit d--cube. This is proved by a bijection which exploits the fact that the h--vector of the simplicial complex arising from the triangulation can be computed via a shelling of the complex. The famous formula P d0 E d x d d! = sec x + tan x, where E d is the number of alternating permutations in S d , is general...

A generalization of the Seidel-Entringer-Arnold method for calculating the alternating permutation numbers (or secant-tangent numbers) leads to a new operation on sequences, the boustrophedon transform. This paper was published (in a somewhat different form) in J. Combinatorial Theory, Series A, 76 (1996), pp. 44–54.

The definitions of descents and excedances in the elements of the symmetric group S d are generalized in two different directions. First, descents and excedances are defined for indexed permutations, i.e. the elements of the group S n d = Z n o S d , where o is wreath product with respect to the usual action of S d by permutation of [d]. It is shown, bijectively, that excedances and descents are equidistributed, and the corresponding descent polynomials, analogous to the Eulerian polynomials, are computed as the feulerian polynomials of simple polynomials. The descent polynomial is shown to equal the h-polynomial (essentially the h-vector) of a certain triangulation of the unit d-cube. This is proved by a bijection which exploits the fact that the h-vector of the triangulation in question can be computed via a shelling of the simplicial complex arising from the triangulation. The h-vector, in turn, is computed via the Ehrhart polynomials of dilations of the unit d-cube. The famous ...

We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. André on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.

We study the distribution of the number of permutations with a given periodic up-down sequence w.r.t. the last entry, nd exponential generating functions and prove asymptotic formulas for this distribution.

We study the number # n of real rational degree n functions (considered up to a linear fractional transformation of the independent variable) with a given set of 2n 2 distinct real critical values. We present a combinatorial interpretation of these numbers and pose a number of related questions.

Abstract. A permutation a1a2 · · · an of 1, 2,..., n is alternating if a1> a2 < a3> a4 < · · ·. We survey some aspects of the theory of alternating permutations, beginning with the famous result of André that if En is the number of alternating permutations of 1, 2,..., n, then P xn n≥0 En = sec x + tan x. n! Topics include refinements and q-analogues of En, various occurrences of En in mathematics, longest alternating subsequences of permutations, umbral enumeration of special classes of alternating permutations, and the connection between alternating permutations and the cd-index of the symmetric group. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday 1. Basic enumerative properties. Let Sn denote the symmetric group of all permutations of [n]: = {1, 2,..., n}. A permutation w = a1a2 · · · an ∈ Sn is called alternating if a1> a2 < a3> a4 < · · ·. In other words, ai < ai+1 for i even, and ai> ai+1 for i odd. Similarly w is reverse alternating if a1 < a2> a3 < a4> · · ·. (Some authors reverse these definitions.) Let En denote the number of alternating permutations in Sn. (Set

. We calculate the number of connected components in the space of the so-called M-polynomials of a given degree in hyperbolic functions. By denition an M-polynomial is characterized by the condition that all its critical values are real and distinct. x1. Introduction The present paper continues the series [1-5] dealing with the topology of the sets of usual and trigonometric polynomials with the maximal possible number of real critical values. Following [1-3], we call a (trigonometric) polynomial with he maximal possible number of real critical values a (trigonometric) M-polynomial. We say that an M-polynomial is generic if all its nite critical values are distinct. We call two rational functions (in particular, two polynomials) f 1 and f 2 equivalent if f 1 = f 2 L, where L is an arbitrary linear fractional transformation of the independent variable. The following results serve as prototypes for the main theorem of this note. Theorem A (see [2]). The number ] pol n of conne...

We study two series of spaces of special real trigonometric polynomials of fixed degree having the maximal possible number of distinct critical values. Those are functions such that either g(' + ß) j \Gammag(') or g(\Gamma') j \Gammag('). For each of the spaces, we calculate the number of its connected components and identify, within the mirror arrangement of the Weyl group of series B, a convex polyhedral model for its closure. A real trigonometric M-polynomial of degree n is one with the maximal number 2n of real critical points. In his recent paper [4], V. I. Arnold constructed a polyhedral model for the manifold of such polynomials and calculated the number of topologically different M-polynomials with all their critical values distinct. The model was provided by a convex cone in the space equipped with the mirror arrangement of the reflection group A 2n\Gamma1 . The enumeration was done in terms of updown sequences (also called A- snakes) of [1, 2]. In the present note we estab...