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 A141052 Number of runs or rising sequences of length 2 among all permutations of n. 0
 1, 4, 21, 130, 930, 7560, 68880, 695520, 7711200, 93139200, 1217462400, 17124307200, 257902444800, 4140968832000, 70614415872000, 1274546617344000, 24275666967552000, 486580401635328000, 10238462617743360000, 225651661258383360000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS Column 2 of A122843. LINKS Persi Diaconis, Mathematical developments from the analysis of riffle shuffling, p. 4. Francis Edward Su, Rising Sequences in Card Shuffling Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society, 1997, pp.120-131. FORMULA a(n) = n!*(5n+1)/4! + floor(2/n)*(1/12), n>=2. Recurrence: a(n) = (n+1)*a(n-1)+(n-1)!/6, n>=2, with a(2)=1 and a(3)=4. E.g.f.: (x^3*(16-11x))/(24(x-1)^2). EXAMPLE a=4 because of the 6 permutations of n=3, there are 4 ascending runs of length 2: {1,3} in {1,3,2} {1,3} in {2,1,3} {2,3} in {2,3,1} {1,2} in {3,1,2} a=4 because of the 6 permutations of n=3, there are 4 rising sequences of length 2: {1,2} in {1,3,2} {2,3} in {2,1,3} {2,3} in {2,3,1} {1,2} in {3,1,2} MATHEMATICA Table[n!(5n + 1)/4! + Floor[2/n](1/12), {n, 2, 10}] CROSSREFS Cf. A122843, A008292, A097900, A001286, A001048, A000142, A028387, A001710. Sequence in context: A232956 A234268 A111177 * A058308 A078591 A090366 Adjacent sequences:  A141049 A141050 A141051 * A141053 A141054 A141055 KEYWORD easy,nonn AUTHOR Harlan J. Brothers, Jul 31 2008, Aug 24 2008 EXTENSIONS First example and typo in second example corrected by Harlan J. Brothers, Apr 29 2013 STATUS approved

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Last modified October 27 19:02 EDT 2021. Contains 348287 sequences. (Running on oeis4.)