Restricted 132-Alternating Permutations and Chebyshev Polynomials

A simple permutation is one that never maps a nontrivial contiguous set of indices contiguously. Given a set of permutations that is closed under taking subpermutations and contains only finitely many simple permutations, we provide a framework for enumerating subsets that are restricted by properties belonging to a finite "query-complete set." Such properties include being even, being an alternating permutation, and avoiding a given generalised (blocked or barred) pattern. We show that the generating functions for these subsets are always algebraic, thereby generalising recent results of Albert and Atkinson. We also apply these techniques to the enumeration of involutions and cyclic closures.

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“…Other results in the literature to which Corollary 1.4 applies appear in [5][6][7][8][9][10][11][12][13][14].…”

Section: Proposition 13 a Union Of Query-complete Sets Of Propertiementioning

confidence: 99%

“…, α m ]-is the permutation obtained by replacing each entry σ (i) by an interval that is order isomorphic to α i . For example, 2413 [1,132,321,12] = 479832156 (see Fig. 2).…”

Section: Introductionmentioning

confidence: 99%

Simple permutations and algebraic generating functions

Brignall

Huczynska

Vatter

2008

Journal of Combinatorial Theory, Series A

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“…(As examples of enumerative results, many papers by T. Mansour and his co-authors on the closely related Tchebyshev polynomials of the second kind come to mind. For a detailed bibliography see [26].) Considering the very symmetric nature of the Tchebyshev posets presented in this paper, one can hope that they may help in many ways to illustrate the properties and understanding of the nature of this extremely important sequence of polynomials.…”

Section: Introductionmentioning

confidence: 97%

Tchebyshev Posets

Hetyei

2004

Discrete Comput Geom

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We construct for each n an Eulerian partially ordered set T n of rank n +1 whose ce-index provides a non-commutative generalization of the nth Tchebyshev polynomial. We show that the order complex of each T n is shellable, homeomorphic to a sphere, and that its face numbers minimize the expression max |x|≤1 | n j=0 ( f j−1 / f n−1 ) · 2 − j · (x − 1) j | among the f -vectors of all (n − 1)-dimensional simplicial complexes. The duals of the posets constructed have a recursive structure similar to face lattices of simplices or cubes, offering the study of a new special class of Eulerian partially ordered sets to test the validity of Stanley's conjecture on the non-negativity of the cd-index of all Gorenstein * posets.

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“…The 231-avoding des-case and its q-analogues. The 231-avoiding alternating permutations were first enumerated by Mansour [32]:…”

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confidence: 99%