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A381539 Number of permutations of [n] having exactly one pair of integers i in [n] such that their cycle minima have opposite sorting order.

%I A381539 #21 Feb 26 2025 21:09:54
%S A381539 0,0,0,1,5,21,89,408,2106,12529,86579,691287,6296333,64454418,
%T A381539 731532528,9101244205,122993164505,1792140943473,27987008885201,
%U A381539 466072881110268,8241676745580774,154187446760870761,3042028843184493887,63114652818792762987,1373581948592359961909
%N A381539 Number of permutations of [n] having exactly one pair of integers i<j in [n] such that their cycle minima have opposite sorting order.
%H A381539 Alois P. Heinz, <a href="/A381539/b381539.txt">Table of n, a(n) for n = 0..300</a>
%H A381539 Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion</a>
%H A381539 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A381539 a(4) = 5: (1)(2,4)(3), (1,2,4)(3), (1,4,2)(3), (1,3)(2)(4), (1,3)(2,4).
%p A381539 b:= proc(o, u, t) option remember; series(`if`(u+o=0, max(0, t-1)!,
%p A381539      `if`(t>0, b(u+o, 0$2)*(t-1)!, 0)+add(x^(u+j-1)*
%p A381539         b(o-j, u+j-1, t+1), j=`if`(t=0, 1, 1..o))), x, 2)
%p A381539     end:
%p A381539 a:= n-> coeff(b(n, 0$2), x, 1):
%p A381539 seq(a(n), n=0..25);
%Y A381539 Column k=1 of A381529.
%K A381539 nonn
%O A381539 0,5
%A A381539 _Alois P. Heinz_, Feb 26 2025