In the present paper we consider the statistic "number of udu's" in Dyck paths. The enumeration of Dyck paths according to semilength and various other parameters has been studied in several papers. However, the statistic "number of udu's" has been considered only recently. Let D n denote the set of Dyck paths of semilength n and let T n, k , L n, k , H n, k and W (r) n, k denote the number of Dyck paths in D n with k udu's, with k udu's at low level, at high level, and at level r ≥ 2, respectively. We derive their generating functions, their recurrence relations and their explicit formulas. A new setting counted by Motzkin numbers is also obtained. Several combinatorial identities are given and other identities are conjectured.

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.

In this note, by the umbra calculus method, the Sun and Zagier's congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.

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Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in set partitions. Let A n,k denote the number of partitions of {1, 2, . . . , n + 1} with the largest singleton {k + 1} for 0 ≤ k ≤ n. In this paper, several explicit formulas for A n,k , involving a Dobinski-type analog, are obtained by algebraic and combinatorial methods, many combinatorial identities involving A n,k and Bell numbers are presented by operator methods, and congruence properties of A n,k are also investigated. It will been showed that the sequences (A n+k,k ) n≥0 and (A n+k,k ) k≥0 (mod p) are periodic for any prime p, and contain a string of p − 1 consecutive zeroes. Moreover their minimum periods are conjectured to be N p = p p −1 p−1 for any prime p.

In this paper we enumerate the number of ways of selecting $k$ objects from $n$ objects arrayed in a line such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects and provide three different formulas when $m,p\geq 1$ and $n\geq pm(k-1)$. Also, we prove that the number of ways of selecting $k$ objects from $n$ objects arrayed in a circle such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects is given by $\frac{n}{n-pk}\binom{n-pk}{k}$, where $m,p\geq 1$ and $n\geq mpk+1$

In this short note, we first present a simple bijection between binary trees and colored ternary trees and then derive a new identity related to generalized Catalan numbers.

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