On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice , spanning a Euclidean space V. Let m be a positive integer and be the arrangement of hyperplanes in V of the form for and . It is known that the number of bounded dominant regions of is equal to the number of facets of the positive part of the generalized cluster complex associated to the pair by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of and conjecture that the corresponding refinement of coincides with the $h$-vector of . We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ, orbits of the action of W on the quotient and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1. 2000 Mathematics Subject Classification Primary—20F55; Secondary—05E99, 20H15     

having a fixed class of nilpotence: combinatorics and enumeration

We study the combinatorics of -nilpotent ideals of a Borel subalgebra of . We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between -nilpotent ideals and Dyck paths. Finally, we propose a -analogue of the Catalan number . These -Catalan numbers count, on the one hand, -nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.

     

Identities for Catalan's Constant Arising from Integrals Depending on a Parameter

In this paper we provide some relationships between Catalan's constant and the ₃F₂ and ₄F₃ hypergeometric functions, deriving them from some parametric integrals. In particular, using the complete elliptic integral of the first kind, we found an alternative proof of a result of Ramanujan for ₃F₂, a second identity related to ₄F₃ and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.     

A Remarkable q,t-Catalan Sequence and q-Lagrange Inversion

We introduce a rational function C _n(q, t) and conjecture that it always evaluates to a polynomial in q, t with non-negative integer coefficients summing to the familiar Catalan number . We give supporting evidence by computing the specializations and C _n (q) = C _n(q,1) = C _n(1,q). We show that, in fact, D _n(q) q-counts Dyck words by the major index and C _n(q) q-counts Dyck paths by area. We also show that C _n(q, t) is the coefficient of the elementary symmetric function e _nin a symmetric polynomial DH_n(x; q, t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that C _n(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DH_n(x; q, 1) yields a novel and combinatorial way of expressing the solution of the q-Lagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis {P (x; q, t)} which are best dealt with in -ring notation. In particular we derive here the -ring version of several symmetric function identities.Work carried out under NSF grant support.     

What power of two divides a weighted Catalan number?

Given a sequence of integers b=(b₀,b₁,b₂,…) one gives a Dyck path P of length 2n the weight

wt(P)=bh₁bh₂?bhn,     

A new statistic on Dyck paths for counting 3-dimensional Catalan words

A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path D, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly D. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.     

Sums of powers of Catalan triangle numbers

In this paper, we consider combinatorial numbers ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$, mentioned as Catalan triangle numbers where $C_{m,k}?\left(\genfrac{}{}{0ex}{}{m?1}{k}\right)?\left(\genfrac{}{}{0ex}{}{m?1}{k?1}\right)$. These numbers unify the entries of the Catalan triangles $B_{n,k}$ and $A_{n,k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n,k}=C_{2n,n?k}$ and $A_{n,k}=C_{2n+1,n+1?k}$. In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers $C_n$ that is $C_{2n,n?1}=C_{2n+1,n}=C_n$.We present identities for sums (and alternating sums) of $C_{m,k}$, squares and cubes of $C_{m,k}$ and, consequently, for $B_{n,k}$ and $A_{n,k}$. In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$ and harmonic numbers ${\left(H_n\right)}_{n\ge 1}$. Finally, in the last section, new open problems and identities involving ${\left(C_n\right)}_{n\ge 0}$ are conjectured.     

Some matrix identities on colored Motzkin paths

Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by using Riordan arrays and a particular model of lattice paths. In this paper, we revisit the identity and emphasize the use of colored partial Motzkin paths as appropriate tool. By using colored Motzkin paths with weight defined according to the height of its last point, we can generalize the identity in several ways. These identities allow us to move from Fibonacci polynomials, Lucas polynomials, and Chebyshev polynomials, to the polynomials of the form ${(z+b)}^n$.