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A368246 Number of permutations of [n] whose cycle minima sum to n.

%I A368246 #22 Jan 04 2024 10:01:33
%S A368246 1,1,0,2,3,8,90,384,2940,18864,232848,1919520,23364000,261282240,
%T A368246 3486637440,48900116160,746747164800,11559784320000,201817271416320,
%U A368246 3580457619916800,68121866659875840,1366946563510886400,28802183294533017600,627950275273991577600
%N A368246 Number of permutations of [n] whose cycle minima sum to n.
%C A368246 Also the number of permutations of [n] for which the sum of the positions of the left-to-right maxima is n: a(4) = 3: 2143, 3142, 3241; a(5) = 8: 31254, 32154, 41253, 41352, 42153, 42351, 43152, 43251.
%H A368246 Alois P. Heinz, <a href="/A368246/b368246.txt">Table of n, a(n) for n = 0..451</a>
%H A368246 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A368246 a(n) = A143946(n,n).
%F A368246 a(n) ~ c * (n-1)!, where c = 0.561459..., conjecture: c = exp(-gamma) = A080130, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Dec 29 2023
%e A368246 a(0) = 1: the empty permutation.
%e A368246 a(1) = 1: (1).
%e A368246 a(2) = 0.
%e A368246 a(3) = 2: (13)(2), (1)(23).
%e A368246 a(4) = 3: (124)(3), (142)(3), (12)(34).
%e A368246 a(5) = 8: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4), (123)(45), (132)(45).
%p A368246 b:= proc(n) option remember; `if`(n=0, 1,
%p A368246       expand(b(n-1)*(x^n+n-1)))
%p A368246     end:
%p A368246 a:= n-> coeff(b(n), x, n):
%p A368246 seq(a(n), n=0..23);
%Y A368246 Main diagonal of A143946.
%Y A368246 Cf. A001620, A080130, A368204, A368678.
%K A368246 nonn
%O A368246 0,4
%A A368246 _Alois P. Heinz_, Dec 18 2023