In this paper, a direct combinatorial proof is given of a result on permutation pairs originally due to Carlitz, Scoville, and Vaughan and later extended. It concerns showing that the series expansion of the reciprocal of a certain multiply exponential generating function has positive integer coefficients. The arguments may then be applied to related problems, one of which concerns the reciprocal of the exponential series for Fibonacci numbers.
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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.
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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.
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