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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, 5198503365971435520000 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
LINKS
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^2*(x-2)*(x-6)/(24*(x-1)^2).
EXAMPLE
a[3]=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[3]=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
Column 2 of A122843.
Sequence in context: A232956 A234268 A111177 * A058308 A078591 A090366
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 March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)