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1
Content available remote

Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

100%
Mező I.
Open Mathematics
|
2013
|
tom 11
|
nr 5
931-939
EN
There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear.
2
Content available remote

A combinatorial proof of a result for permutation pairs

81%
Mansour T., Shattuck M.
Open Mathematics
|
2012
|
tom 10
|
nr 2
797-806
EN
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.
3
Content available remote

Parity-alternating permutations and successions

81%
Munagi A.
Open Mathematics
|
2014
|
tom 12
|
nr 9
1390-1402
EN
The study of parity-alternating permutations of {1, 2, … n} is extended to permutations containing a prescribed number of parity successions - adjacent pairs of elements of the same parity. Several enumeration formulae are computed for permutations containing a given number of parity successions, in conjunction with further parity and length restrictions. The objects are classified using direct construction and elementary combinatorial techniques. Analogous results are derived for circular permutations.
4
Content available remote

Restricted partitions and q-Pell numbers

81%
Mansour T., Shattuck M.
Open Mathematics
|
2011
|
tom 9
|
nr 2
346-355
EN
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.
5
Content available remote

Plane trivalent trees and their patterns

81%
Delorme C.
Open Mathematics
|
2010
|
tom 8
|
nr 6
1041-1047
EN
The aim of this paper is to characterize the patterns of successive distances of leaves in plane trivalent trees, and give a very short characterization of their parity pattern. Besides, we count how many trees satisfy some given sequences of patterns.
6
Content available remote

Pattern avoiding partitions and Motzkin left factors

81%
Mansour T., Shattuck M.
Open Mathematics
|
2011
|
tom 9
|
nr 5
1121-1134
EN
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.
7
Content available remote

Secant tree calculus

62%
Foata D., Han G.-N.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1852-1870
EN
A true Tree Calculus is being developed to make a joint study of the two statistics “eoc” (end of minimal chain) and “pom” (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.
8
Content available remote

Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

62%
Blecher A., Brennan C., Mansour T.
Open Mathematics
|
2012
|
tom 10
|
nr 2
788-796
EN
Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”. From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.
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