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

1, 1, 0, 2, 3, 8, 90, 384, 2940, 18864, 232848, 1919520, 23364000, 261282240, 3486637440, 48900116160, 746747164800, 11559784320000, 201817271416320, 3580457619916800, 68121866659875840, 1366946563510886400, 28802183294533017600, 627950275273991577600

Offset: 0

Author

Alois P. Heinz, Dec 18 2023

Keywords

• nonn

Comments

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.

Examples

			a(0) = 1: the empty permutation.
a(1) = 1: (1).
a(2) = 0.
a(3) = 2: (13)(2), (1)(23).
a(4) = 3: (124)(3), (142)(3), (12)(34).
a(5) = 8: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4), (123)(45), (132)(45).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..451
• Wikipedia, Permutation

Crossrefs

Main diagonal of A143946.

Cf. A001620, A080130, A368204, A368678.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      expand(b(n-1)*(x^n+n-1)))
    end:
a:= n-> coeff(b(n), x, n):
seq(a(n), n=0..23);

Formula

a(n) = A143946(n,n).

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