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

Author

Alois P. Heinz, Aug 29 2013

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Column k=2 of A064315.

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

Programs

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 *)

Formula

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