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1
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A combinatorial proof of a result for permutation pairs

100%
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.
2
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Parity-alternating permutations and successions

100%
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.
3
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The visibility parameter for words and permutations

100%
Cristea L., Prodinger H.
Open Mathematics
|
2013
|
tom 11
|
nr 2
283-295
EN
We investigate the visibility parameter, i.e., the number of visible pairs, first for words over a finite alphabet, then for permutations of the finite set {1, 2, …, n}, and finally for words over an infinite alphabet whose letters occur with geometric probabilities. The results obtained for permutations correct the formula for the expectation obtained in a recent paper by Gutin et al. [Gutin G., Mansour T., Severini S., A characterization of horizontal visibility graphs and combinatorics on words, Phys. A, 2011, 390 (12), 2421–2428], and for words over a finite alphabet the formula obtained in the present paper for the expectation is more precise than that obtained in the cited paper. More importantly, we also compute the variance for each case.
4
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A variant of the reciprocal super Catalan matrix

100%
Kılıç E., Akkus I., Kızılaslan G.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
Recently Prodinger [8] considered the reciprocal super Catalan matrix and gave explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and obtained some related matrices. For all results, q-analogues were also presented. In this paper, we define and study a variant of the reciprocal super Catalan matrix with two additional parameters. Explicit formulæ for its LU-decomposition, LUdecomposition of its inverse and the Cholesky decomposition are obtained. For all results, q-analogues are also presented.
5
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The reciprocal super Catalan matrix

100%
Prodinger H.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
The reciprocal super Catalan matrix has entries [...] . Explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, q-analogues are also presented.
6
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The box parameter for words and permutations

100%
Prodinger H.
Open Mathematics
|
2014
|
tom 12
|
nr 1
167-174
EN
The box parameter for words counts how often two letters w j and w k define a “box” such that all the letters w j+1; ..., w k−1 fall into that box. It is related to the visibility parameter and other parameters on words. Three models are considered: Words over a finite alphabet, permutations, and words with letters following a geometric distribution. A typical result is: The average box parameter for words over an M letter alphabet is asymptotically given by 2n − 2n H M/M, for fixed M and n → ∞.
7
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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.
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