A238978 Number of ballot sequences of length n with exactly 3 fixed points.

0, 0, 0, 1, 1, 3, 9, 28, 93, 321, 1168, 4404, 17328, 70408, 296436, 1284768, 5740804, 26332788, 124066608, 598625296, 2958281328, 14941136784, 77111251408, 406028059968, 2180584156176, 11930067296848, 66468429865344, 376770132276288, 2172036623279488

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Mar 07 2014

Keywords

• nonn
• easy

Comments

The fixed points are in the first 3 positions.

Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, and 3, but not 4. An alternate definition uses the first row.

Examples

			a(3) = 1: [1,2,3].
a(4) = 1: [1,2,3,1].
a(5) = 3: [1,2,3,1,1], [1,2,3,1,2], [1,2,3,1,4].
a(6) = 9: [1,2,3,1,1,1], [1,2,3,1,1,2], [1,2,3,1,1,4], [1,2,3,1,2,1], [1,2,3,1,2,3], [1,2,3,1,2,4], [1,2,3,1,4,1], [1,2,3,1,4,2], [1,2,3,1,4,5].
		

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..800
• Wikipedia, Young tableau

Crossrefs

Column k=3 of A238802.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, n*(n-1)*(n-2)/6,
      ((4*n^3-54*n^2+216*n-254) *a(n-1)
       +(n-5)*(3*n^3-31*n^2+84*n-30) *a(n-2)
       -(n-5)*(n-6)*(n^2-3*n-8) *a(n-3)) /
      ((n-3)*(3*n^2-33*n+86)))
    end:
seq(a(n), n=0..40);

Mathematica

b[n_, l_List] := b[n, l] = If[n <= 0, 1, b[n - 1, Append[l, 1]] + Sum[If[i == 1 || l[[i - 1]] > l[[i]], b[n - 1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; a[n_] := If[n == 3, 1, b[n - 4, {2, 1, 1}]]; a[n_ /; n < 3] = 0; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Formula

See Maple program.

a(n) ~ sqrt(2)/16 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014

Recurrence (for n>=5): (n-4)*(n^3 - 10*n^2 + 27*n - 26)*a(n) = (n^4 - 14*n^3 + 67*n^2 - 150*n + 152)*a(n-1) + (n-5)*(n-3)*(n^3 - 7*n^2 + 10*n - 8)*a(n-2). - Vaclav Kotesovec, Mar 08 2014