A185652 Number of permutations of [n] having a shortest ascending run of length 2.

0, 0, 1, 0, 5, 18, 89, 519, 3853, 27555, 233431, 2167152, 21596120, 232817282, 2718706924, 33814848445, 448311181346, 6319365554730, 94225534689624, 1481940898130323, 24536143182460549, 426432943716156580, 7762187693343502658, 147704506384475066381

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

Formula

a(n) ~ c * (3*sqrt(3)/(2*Pi))^n * n!, where c = 0.45178068752734823... . - Vaclav Kotesovec, Sep 06 2014

Example

a(2) = 1: 12.
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 18: 12435, 12534, 13245, 13425, 13524, 14235, 14523, 15234, 23145, 23415, 23514, 24135, 24513, 25134, 34125, 34512, 35124, 45123.

Mathematica

A[n_, k_] := A[n, k] = Module[{b}, b[u_, o_, t_] := b[u, o, t] = If[t + o <= k, (u + o)!, Sum[b[u + i - 1, o - i, Min[k, t] + 1], {i, 1, o}] + If[t <= k, u (u + o - 1)!, Sum[b[u - i, o + i - 1, 1], {i, 1, u}]]]; Sum[b[j - 1, n - j, 1], {j, 1, n}]];
a[n_] := A[n, 2] - A[n, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 26 2021, after Alois P. Heinz in A064315 *)

Crossrefs

Column k=2 of A064315.

Cf. A086089 (3*sqrt(3)/(2*Pi)).

Sequence in context: A213190 A151490 A282475 * A009305 A097574 A113023

Adjacent sequences: A185649 A185650 A185651 * A185653 A185654 A185655

Keyword

nonn

Author

Alois P. Heinz, Aug 29 2013

Status

approved