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

0,4

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.

Links

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

Wikipedia, Permutation

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

Example

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).

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);

Crossrefs

Main diagonal of A143946.

Cf. A001620, A080130, A368204, A368678.

Sequence in context: A191353 A319218 A358562 * A243954 A005008 A079938

Adjacent sequences: A368243 A368244 A368245 * A368247 A368248 A368249

Keyword

nonn

Author

Alois P. Heinz, Dec 18 2023

Status

approved