A238979 Number of ballot sequences of length n with exactly 4 fixed points.

0, 0, 0, 0, 1, 1, 3, 9, 29, 98, 350, 1302, 5062, 20380, 85108, 366444, 1627836, 7430360, 34855016, 167546408, 825185448, 4155400720, 21388745008, 112355110672, 602103194448, 3287743832352, 18285157048544, 103480813034336, 595671084096608, 3485006638408128

Offset

0,7

Comments

The fixed points are in the first 4 positions.

Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, 3, and 4, but not 5. An alternate definition uses the first row.

Conjecture: Generally, for fixed k is column k of A238802 asymptotic to sqrt(2)/(2*(k+1)*(k-1)!) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))), holds for all k<=10. - Vaclav Kotesovec, Mar 08 2014

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)/60 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014

Recurrence (for n>=6): (n-5)*(n^4 - 15*n^3 + 65*n^2 - 45*n + 54)*a(n) = (n^5 - 20*n^4 + 125*n^3 - 145*n^2 - 651*n + 810)*a(n-1) + (n-6)*(n-4)*(n^4 - 11*n^3 + 26*n^2 + 44*n + 60)*a(n-2). - Vaclav Kotesovec, Mar 08 2014

Example

a(4) = 1: [1,2,3,4].
a(5) = 1: [1,2,3,4,1].
a(6) = 3: [1,2,3,4,1,1], [1,2,3,4,1,2], [1,2,3,4,1,5].
a(7) = 9: [1,2,3,4,1,1,1], [1,2,3,4,1,1,2], [1,2,3,4,1,1,5], [1,2,3,4,1,2,1], [1,2,3,4,1,2,3], [1,2,3,4,1,2,5], [1,2,3,4,1,5,1], [1,2,3,4,1,5,2], [1,2,3,4,1,5,6].

Maple

b:= proc(n) option remember; `if`(n<3, [1$2, 3][n+1],
      ((11-n)*b(n-1) +(n^3+4*n^2-15)*b(n-2)
       +(n-1)*(n-3)*(n+7)*b(n-3))/((n-1)*(n+1)))
    end:
a:= n-> `if`(n<4, 0, b(n-4)):
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 == 4, 1, b[n - 5, {2, 1, 1, 1}]]; a[n_ /; n < 4] = 0; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Crossrefs

Column k=4 of A238802.

Sequence in context: A346158 A077587 A001893 * A151030 A066331 A099780

Adjacent sequences: A238976 A238977 A238978 * A238980 A238981 A238982

Keyword

nonn,easy

Author

Joerg Arndt and Alois P. Heinz, Mar 07 2014

Status

approved