zbMATH — the first resource for mathematics

On the exact distributions of Eulerian and Simon Newcomb numbers associated with random permutations. (English) Zbl 1057.62503
Summary: Eulerian and Simon Newcomb numbers are two of the most celebrated numbers associated with random permutations. Their distributions have been successfully used in various areas of statistics and applied probability. Conventionally, these distributions have been studied via combinatorial analysis. We provide a new, simple and unified probabilistic method based on the finite Markov chain imbedding technique to study the exact distributions of Eulerian and Simon Newcomb numbers. A new recursive equation which characterizes the Simon Newcomb numbers is obtained. We also show that many classical identities and recursive equations associated with Eulerian numbers are immediate consequences of our main result.

MSC:
 62E15 Exact distribution theory in statistics 11B68 Bernoulli and Euler numbers and polynomials
Full Text:
References:
 [1] Carlitz, L., Extended Bernoulli and Eulerian numbers, J. duke math., 31, 667-690, (1964) · Zbl 0127.29501 [2] Carlitz, L., 1972. Enumeration of sequences by rises and falls: A refinement of the Simon Newcomb problem. J. Duke Math. 39, 267-280. · Zbl 0243.05008 [3] Carlitz, L., Permutations and sequences, Adv. math., 14, 92-120, (1974) · Zbl 0285.05011 [4] Carlitz, L.; Scoville, R.A., Generalized Eulerian numbers: combinatorial applications, Journal für die reine und angewandte Mathematik, 265, 110-137, (1974) · Zbl 0276.05006 [5] David, F.N., Barton, D.E., 1962. Combinatorial Chance. Hafiner, New York. [6] Dillon, J.F.; Roselle, D., Simon Newcomb’s problem, SIAM J. appl. math., 17, 1086-1093, (1969) · Zbl 0212.34701 [7] Dwyer, P.S., The cumulative numbers and their polynomials, Ann. math. statist., 11, 66-71, (1940) · Zbl 0063.01200 [8] Fu, J.C., Reliability of consecutive-k-out-of-n: F systems with (k−1)-step Markov dependence, IEEE trans. reliab., R35, 602-606, (1986) · Zbl 0612.60079 [9] Fu, J.C., Exact and limiting distributions of the number of successions in a random permutation, Ann. inst. statist. math., 47, 435-446, (1995) · Zbl 0841.60007 [10] Fu, J.C., Distribution theory of runs and patterns associated with a sequence of multi-state trials, Statistica sinica, 6, 957-974, (1996) · Zbl 0857.60068 [11] Fu, J.C.; Koutras, M.V., Distribution theory of runs: a Markov chain approach, J. amer. statist. assoc., 89, 1050-1058, (1994) · Zbl 0806.60011 [12] Giladi, E.; Keller, J.B., Eulerian number asymptotics, Proc. roy. soc. lond. A, 445, 291-303, (1994) · Zbl 0837.05012 [13] Harris, B.; Park, C.J., A generalization of the Eulerian numbers with a probabilistic application, Statist. probab. lett., 20, 37-47, (1994) · Zbl 0801.60013 [14] Koutras, M.V., Eulerian numbers associated with sequences of polynomials, Fibonacci quart., 32, 44-57, (1994) · Zbl 0791.05002 [15] Koutras, M.V.; Alexandrou, V., Runs, scans and urn model distributions: a unified Markov chain approach, Ann. inst. statist. math., 47, 743-766, (1995) · Zbl 0848.60021 [16] Lou, W.Y.W., On runs and longest run tests: a method of finite Markov chain imbedding, J. amer. statist. assoc., 91, 1595-1601, (1996) · Zbl 0881.62086 [17] MacMahon, P.A., 1915. Combinatory Analysis. Cambridge, London. · JFM 45.1271.01 [18] Nicolas, J.L., An integral representation for Eulerian numbers, Colloq. math. soc., 60, 513-527, (1992) · Zbl 0794.05005 [19] Riordan, J., 1958. An Introduction to Combinatorial Analysis. Wiley, New York. · Zbl 0078.00805 [20] Roselle, D., Permutations by number rises and successions, Proc. A.M.S, 19, 8-16, (1969) · Zbl 0159.30101 [21] Takacs, L., A generalization of the Eulerian numbers, Publ. math. debrecen, 26, 173-181, (1979) · Zbl 0442.05001 [22] Tanny, S., A probabilistic interpretation of Eulerian numbers, J. duke math., 40, 717-722, (1973) · Zbl 0284.05006 [23] Worpitzky, J., Studien über die bernoullischen und eulerischen zahlen, J. reine angew. math., 94, 203-232, (1883) · JFM 15.0201.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.