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1
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Restricted partitions and q-Pell numbers

100%
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.
2
Content available remote

On colored set partitions of type B n

100%
Wang D.
Open Mathematics
|
2014
|
tom 12
|
nr 9
1372-1381
EN
Generalizing Reiner’s notion of set partitions of type B n, we define colored B n-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored B n-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored B n-partition. We find an asymptotic expression of the total number of colored B n-partitions up to an error of O(n −1/2log7/2 n], and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored B n-partitions.
3
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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
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.
4
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Permutations which make transitive groups primitive

76%
Lopes P.
Open Mathematics
|
2009
|
tom 7
|
nr 4
650-659
EN
In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and certain subgroups of some affine special linear groups.
5
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Differential equations forp,q-Touchard polynomials

64%
Kim T., Herscovici O., Mansour T., Rim S.-H.
Open Mathematics
|
2016
|
tom 14
|
nr 1
908-912
EN
In this paper, we present differential equation for the generating function of the p, q-Touchard polynomials. An application to ordered partitions of a set is investigated.
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