Combinatorial proofs of some Moriarty-type binomial coefficient identities.
Combinatorial proofs of some Simons-type binomial coefficient identities.
Parity theorems for statistics on permutations and Catalan words.
Bijective proofs of parity theorems for partition statistics.
Tiling proofs of some Fibonacci-Lucas relations.
Tiling proofs of some formulas for the Pell numbers of odd index.
Counting peaks and valleys in a partition of a set.
Parity theorems for statistics on domino arrangements.
Periodicity and parity theorems for a statistic on -mino arrangements.
Parity theorems for statistics on lattice paths and Laguerre configurations.
Some generalized Fibonacci polynomials.
A new statistic on linear and circular -mino arrangements.
Recurrence relations and two-dimensional set partitions.
Recounting determinants for a class of Hessenberg matrices.
On a new family of generalized Stirling and Bell numbers.
Compression theorems for periodic tilings and consequences.
Counting subwords in a partition of a set.
Restricted partitions and q-Pell numbers
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...
A combinatorial proof of a result for permutation pairs
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.
Pattern avoiding partitions and Motzkin left factors
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...
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