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A216119
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Number of stretching pairs in all permutations in S_n.
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3
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0, 0, 0, 2, 30, 360, 4200, 50400, 635040, 8467200, 119750400, 1796256000, 28540512000, 479480601600, 8499883392000, 158664489984000, 3112264995840000, 64023737057280000, 1378644471300096000, 31019500604252160000, 728045925946859520000, 17796678189812121600000
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OFFSET
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1,4
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COMMENTS
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A stretching pair of a permutation p in S_n is a pair (i,j) (1 <= i < j <= n) satisfying p(i) < i < j < p(j). For example, for the permutation 31254 in S_5 the pair (2,4) is stretching because p(2) = 1 < 2 < 4 < p(4) = 5.
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REFERENCES
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E. Lundberg and B. Nagle, A permutation statistic arising in dynamics of internal maps. (submitted)
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LINKS
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FORMULA
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a(n) = n!*(n-2)*(n-3)/24.
Sum_{n>=4} 1/a(n) = 8*(gamma - Ei(1)) + 8*e - 32/3, where gamma = A001620, Ei(1) = A091725, and e = A001113.
Sum_{n>=4} (-1)^n/a(n) = 16*(gamma - Ei(-1)) - 8/e - 28/3, where Ei(-1) = -A099285. (End)
D-finite with recurrence a(n) +(-n-10)*a(n-1) +4*(2*n+3)*a(n-2) +12*(-n+2)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
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EXAMPLE
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a(4) = 2 because 2143 has 1 stretching (namely (2,3)), 3142 has 1 stretching pair (namely (2,3)), and the other 22 permutations in S_4 have no stretching pairs.
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MAPLE
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0, seq((1/24)*factorial(n)*(n-2)*(n-3), n = 2 .. 22);
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MATHEMATICA
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Join[{0}, Table[n! (n - 2) (n - 3) / 24, {n, 2, 30}]] (* Vincenzo Librandi, Nov 29 2018 *)
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PROG
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(Magma) [Factorial(n)*(n-2)*(n-3) div 24: n in [1..30]]; // Vincenzo Librandi, Nov 29 2018
(GAP) Concatenation([0], List([2..22], n->Factorial(n)*(n-2)*(n-3)/24)); # Muniru A Asiru, Nov 29 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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