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In this paper we show that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers are Hausdorff moment sequences in a unified approach. In particular we answer a conjecture of Liang at al. which such numbers have unique representing measures. The smallest interval including the support of representing measure is explicitly found. Subsequences of Catalan-like numbers are also considered. We provide a necessary and sufficient condition for a pattern of subsequences that if sequences are the Stieltjes Catalan-like numbers, then their subsequences are Stieltjes Catalan-like numbers. Moreover, a representing measure of a linear combination of consecutive Catalan-like numbers is studied.
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We study integral representation of so-called $d$-dimensional Catalan numbers $C_{d}(n)$, defined by $[prod_{p=0}^{d-1} frac{p!}{(n+p)!}] (d n)!$, $d = 2, 3, ...$, $n=0, 1, ...$. We prove that the $C_{d}(n)$s are the $n$th Hausdorff power moments of positive functions $W_{d}(x)$ defined on $xin[0, d^d]$. We construct exact and explicit forms of $W_{d}(x)$ and demonstrate that they can be expressed as combinations of $d-1$ hypergeometric functions of type $_{d-1}F_{d-2}$ of argument $x/d^d$. These solutions are unique. We analyse them analytically and graphically. A combinatorially relevant, specific extension of $C_{d}(n)$ for $d$ even in the form $D_{d}(n)=[prod_{p = 0}^{d-1} frac{p!}{(n+p)!}] [prod_{q = 0}^{d/2 - 1} frac{(2 n + 2 q)!}{(2 q)!}]$ is analyzed along the same lines.
Given a permutation $f$, we study the positroid Catalan number $C_f$ defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated $q,t$-polynomials coincide with the generalized $q,t$-Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov-Rozansky homology of Coxeter links.
The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the nth Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we propose a combinatorial formula for a q-analogue of the rational Catalan numbers and relate this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q,t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.
We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.
We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $mathbb{C}$ and point counts over $mathbb{F}_q$) to Khovanov--Rozansky homology of associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational $q,t$-Catalan numbers. Via the curious Lefschetz property of cluster varieties, this implies the $q,t$-symmetry and unimodality properties of rational $q,t$-Catalan numbers. We show that the $q,t$-symmetry phenomenon is a manifestation of Koszul duality for category $mathcal{O}$, and discuss relations with open Richardson varieties and extension groups of Verma modules.
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