OFFSET
1,4
COMMENTS
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.
REFERENCES
E. Lundberg and B. Nagle, A permutation statistic arising in dynamics of internal maps. (submitted)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..450
E. Clark and R. Ehrenborg, Explicit expressions for the extremal excedance statistic, European J. Combinatorics, 31 (2010), 270-279.
J. Cooper, E. Lundberg, and B. Nagle, Generalized pattern frequency in large permutations, Electron. J. Combin., 20, (2013), Article P28.
FORMULA
a(n) = n!*(n-2)*(n-3)/24.
a(n) = 2*A005461(n-3).
a(n) = Sum_{k>=1} A216118(k).
a(n) = Sum_{k>=1} k*A216120(n,k).
From Amiram Eldar, May 06 2022: (Start)
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
EXAMPLE
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.
MAPLE
0, seq((1/24)*factorial(n)*(n-2)*(n-3), n = 2 .. 22);
MATHEMATICA
Join[{0}, Table[n! (n - 2) (n - 3) / 24, {n, 2, 30}]] (* Vincenzo Librandi, Nov 29 2018 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 26 2013
STATUS
approved