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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.
2
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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 → ∞.
3
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Secant tree calculus

76%
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.
4
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On the k-gamma q-distribution

76%
Díaz R., Ortiz C., Pariguan E.
Open Mathematics
|
2010
|
tom 8
|
nr 3
448-458
EN
We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Díaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.
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