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In this paper, we present a reduction algorithm which transforms m-regular partitions of [n] = {1, 2,. .. , n} to (m − 1)-regular partitions of [n − 1]. We show that this algorithm preserves the noncrossing property. This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing… (More)

We obtain a characterization of (321, 3 ¯ 142)-avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of (321, 3 ¯ 142)-avoiding permutations of length n equals the n-th Motzkin number, due to Gire, and further studied by Barcucci, Del Lungo,… (More)

This paper is devoted to characterize permutations with forbidden patterns by using canonical reduced decompositions, which leads to bijections between Dyck paths and S n (321) and S n (231), respectively. We also discuss permutations in S n avoiding two patterns, one of length 3 and the other of length k. These permutations produce a kind of discrete… (More)

Riordan paths are Motzkin paths without horizontal steps on the x-axis. We establish a correspondence between Riordan paths and (321, 3 ¯ 142)-avoiding derangements. We also present a combinatorial proof of a recurrence relation for the Riordan numbers in the spirit of the Foata-Zeilberger proof of a recurrence relation on the Schröder numbers.

- Li-Hua Deng, Eva Y. P. Deng, Louis W. Shapiro
- 2009

In this paper, we study symmetric lattice paths. Let d n , m n , and s n denote the number of symmetric Dyck paths, symmetric Motzkin paths, and symmetric Schröder paths of length 2n, respectively. By using Riordan group methods we obtain six identities relating d n , m n , and s n and also give two of them combinatorial proofs. Finally, we investigate some… (More)

Let A k be the set of permutations in the symmetric group S k with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns A k. We present a bijection between symmetric Schröder paths of length 2n and involutions of length n + 1 avoiding A4. Statistics such as the number of right-to-left maxima and fixed points of the… (More)

In this paper we solve several recurrence relations with two (three) indices using combinatorial methods. Moreover, we present several recurrence relations with two (three) indices related to ternary paths and k-ary paths.

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