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Pattern avoiding partitions and Motzkin left factors

100%

Mansour T., Shattuck M.

Open Mathematics

2011

tom 9

nr 5

1121-1134

Let L n, n ≥ 1, denote the sequence which counts the number of paths from the origin to the line x = n − 1 using (1, 1), (1, −1), and (1, 0) steps that never dip below the x-axis (called Motzkin left factors). The numbers L n count, among other things, certain restricted subsets of permutations and Catalan paths. In this paper, we provide new combinatorial interpretations for these numbers in terms of finite set partitions. In particular, we identify four classes of the partitions of size n, all of which have cardinality L n and each avoiding a set of two classical patterns of length four. We obtain a further generalization in one of the cases by considering a pair of statistics on the partition class. In a couple of cases, to show the result, we make use of the kernel method to solve a functional equation arising after a certain parameter has been introduced.

Restricted partitions and q-Pell numbers

88%

Mansour T., Shattuck M.

Open Mathematics

2011

tom 9

nr 2

346-355

In this paper, we provide new combinatorial interpretations for the Pell numbers p n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p n. By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p n. Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.

Ograniczanie wyników

2 Open Mathematics

2 Mansour T.

2 Shattuck M.

2 2011

Wyniki wyszukiwania

Pattern avoiding partitions and Motzkin left factors

Restricted partitions and q-Pell numbers