The visibility parameter for words and permutations

• Autorzy: Ligia Cristea , Helmut Prodinger
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Words; Permutations; q-enumeration

Kategorie tematyczne

• 05A05: Permutations, words, matrices
• 05A30: q -calculus and related topics

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 2
• Strony: 283-295

Daty

wydano

2013-02-01

online

2012-11-21

Twórcy

autor

Ligia Cristea

• Technische Universität Graz

autor

Helmut Prodinger

• Stellenbosch University

Bibliografia

• [1] Flajolet Ph., Martin G.N., Probabilistic counting algorithms for data base applications, J. Comput. System Sci., 1985, 31(2), 182–209 http://dx.doi.org/10.1016/0022-0000(85)90041-8
• [2] Graham R.L., Knuth D.E., Patashnik O., Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, 1994
• [3] Gutin G., Mansour T., Severini S., A characterization of horizontal visibility graphs and combinatorics on words, Phys. A, 2011, 390(12), 2421–2428 http://dx.doi.org/10.1016/j.physa.2011.02.031
• [4] Prodinger H., Combinatorics of geometrically distributed random variables: left-to-right maxima, In: 5th Conference on Formal Power Series and Algebraic Combinatorics, Florence, June 21–25, 1993, Discrete Math., Elsevier, Amsterdam, 1996, 153(1–3), 253–270
• [5] Prodinger H., A q-analogue of the path length of binary search trees, Algorithmica, 2001, 31(3), 433–441 http://dx.doi.org/10.1007/s00453-001-0058-y
• [6] Prodinger H., Combinatorics of geometrically distributed random variables: inversions and a parameter of Knuth, Ann. Comb., 2001, 5(2), 241–250 http://dx.doi.org/10.1007/s00026-001-8010-z
• [7] Pugh W., Skip lists: a probabilistic alternative to balanced trees, Communications of the ACM, 1990, 33(6), 668–676 http://dx.doi.org/10.1145/78973.78977

Identyfikatory

DOI

10.2478/s11533-012-0135-2