Some q-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths

Bonin, Joseph E.; Shapiro, Louis W.; Simion, Rodica

doi:10.1016/0378-3758(93)90032-2

Cited by 104 publications

(114 citation statements)

References 14 publications

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“…There are several ways to define q-analogues of the Schröder numbers (see [1,2,5]). We will need the simplest one (see [5], page 37, polynomials d n (q)).…”

Section: Poincaré Polynomialsmentioning

confidence: 99%

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Degenerate flag varieties: moment graphs and Schröder numbers

2012

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We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T . Using the group action, we describe the moment graphs, encoding the zero-and one-dimensional T -orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal steps in a Schröder path. As an application we obtain a new combinatorial description of the large and small Schröder numbers and their q-analogues.

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“…There are several ways to define q-analogues of the Schröder numbers (see [1,2,5]). We will need the simplest one (see [5], page 37, polynomials d n (q)).…”

Section: Poincaré Polynomialsmentioning

confidence: 99%

“…We will need the simplest one (see [5], page 37, polynomials d n (q)). They are called Narayana polynomials there, but in other papers the same polynomials are also referred to as Schröder polynomials, see e.g.…”

Section: Poincaré Polynomialsmentioning

confidence: 99%

Degenerate flag varieties: moment graphs and Schröder numbers

2012

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“…. are in bijection with Schrörder paths, which have been studied in [1] and [12]. In this section, we derive explicit formulas for ( We use the Appell relation…”

Section: Explicit Formulas For (1 B)-pathsmentioning

confidence: 99%

Lattice and Schröder paths with periodic boundaries

Kung

Mier

Sun

et al. 2009

Journal of Statistical Planning and Inference

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“…The systematic study of noncrossing partitions began with Kreweras [7] and Poupard [10]. For some further work on noncrossing partitions, see [2], [3], [5], [6], [9], [10], [11], [12], [13], [14] and the references given there. Let f (n 1 , n 2 , · · · , n p ) denote the number of noncrossing partitions of [n] into p parts of given sizes n 1 , n 2 , · · · , n p (but not specifying which part gets which size); and let p k denote the number of parts with size k. Kreweras [7] gave the beautiful and surprising result (also see [4]):…”

Section: Introductionmentioning

confidence: 99%

A simple and direct derivation for the number of noncrossing partitions

Liaw¹,

Yeh²,

Hwang³

et al. 1998

Proc. Amer. Math. Soc.

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Abstract. Kreweras considered the problem of counting noncrossing partitions of the set {1, 2, · · · , n}, whose elements are arranged into a cycle in its natural order, into p parts of given sizes n 1 , n 2 , · · · , np (but not specifying which part gets which size). He gave a beautiful and surprising result whose proof resorts to a recurrence relation. In this paper we give a direct, entirely bijective, proof starting from the same initial idea as Kreweras' proof.

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Some q-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths

Cited by 104 publications

References 14 publications

Degenerate flag varieties: moment graphs and Schröder numbers

Degenerate flag varieties: moment graphs and Schröder numbers

Lattice and Schröder paths with periodic boundaries

A simple and direct derivation for the number of noncrossing partitions

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