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

0,6

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.

Links

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

Wikipedia, Young tableau

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

Example

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].

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 *)

Crossrefs

Column k=3 of A238802.

Sequence in context: A131203 A191637 A345241 * A081914 A361763 A120985

Adjacent sequences: A238975 A238976 A238977 * A238979 A238980 A238981

Keyword

nonn,easy

Author

Joerg Arndt and Alois P. Heinz, Mar 07 2014

Status

approved