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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine 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 ..."
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Cited by 631 (15 self)
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This article gives a brief introduction to the OnLine 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
Permutation Statistics of Indexed Permutations
, 1994
"... 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 p ..."
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Cited by 38 (2 self)
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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 feulerian polynomial of a simple polynomial. The descent polynomial is shown to equal the hpolynomial (essentially the hvector) of a certain triangulation of the unit dcube. This is proved by a bijection which exploits the fact that the hvector 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 new operation on sequences: the boustrophedon transform
 J. Combin. Th. Ser. A
, 1996
"... A generalization of the SeidelEntringerArnold method for calculating the alternating permutation numbers (or secanttangent 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 ..."
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Cited by 12 (0 self)
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A generalization of the SeidelEntringerArnold method for calculating the alternating permutation numbers (or secanttangent 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.
Permutation Statistics of Indexed and Poset Permutations
"... 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 u ..."
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Cited by 7 (3 self)
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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 hpolynomial (essentially the hvector) of a certain triangulation of the unit dcube. This is proved by a bijection which exploits the fact that the hvector of the triangulation in question can be computed via a shelling of the simplicial complex arising from the triangulation. The hvector, in turn, is computed via the Ehrhart polynomials of dilations of the unit dcube. The famous ...
Enumeration formulas for Young tableaux in a diagonal strip
 Israel J. Math
"... 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 updown permutations. The analysis uses a transfer operator approach extending the me ..."
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Cited by 6 (0 self)
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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 updown 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.
A Survey of Alternating Permutations
, 2009
"... 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 ..."
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Cited by 6 (1 self)
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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 qanalogues 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 cdindex 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
Periodic De Bruijn Triangles: Exact and Asymptotic Results
 Discrete Math
, 2003
"... We study the distribution of the number of permutations with a given periodic updown sequence w.r.t. the last entry, nd exponential generating functions and prove asymptotic formulas for this distribution. ..."
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Cited by 3 (0 self)
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We study the distribution of the number of permutations with a given periodic updown sequence w.r.t. the last entry, nd exponential generating functions and prove asymptotic formulas for this distribution.
Counting Real Rational Functions With All Real Critical Values
"... 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 quest ..."
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Cited by 3 (1 self)
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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.
ON THE BASIS POLYNOMIALS IN THE THEORY OF PERMUTATIONS WITH PRESCRIBED UPDOWN STRUCTURE
, 801
"... Abstract. We study the polynomials which enumerate the permutations π = (π1, π2,..., πn) of the elements 1.2,..., n with the condition π1 < π2 <... < πn−m(or π1> π2>...> πn−m) and prescribed updown points n −m, n −m+1,..., n − 1 in view of an important role of these polynomials in theory of enumera ..."
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Cited by 2 (2 self)
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Abstract. We study the polynomials which enumerate the permutations π = (π1, π2,..., πn) of the elements 1.2,..., n with the condition π1 < π2 <... < πn−m(or π1> π2>...> πn−m) and prescribed updown points n −m, n −m+1,..., n − 1 in view of an important role of these polynomials in theory of enumeration the permutations with prescribed updown structure similar to the role of the binomial coefficients in the enumeration of the subsets of a finite set satisfying some restrictions. 1.
Möbius and odd real trigonometric Mfunctions
, 1997
"... 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 conne ..."
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Cited by 1 (0 self)
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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 Mpolynomial 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 Mpolynomials 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...