A239118 Number of ballot sequences of length n with exactly 7 fixed points.

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 9, 29, 99, 357, 1351, 5342, 21983, 93823, 414198, 1886424, 8846390, 42628782, 210773592, 1067599984, 5533263752, 29307314408, 158484944416, 874103230896, 4913196556800, 28120097476640, 163770757573776, 969858742317600

Offset

0,10

Comments

The fixed points are in the first 7 positions.

Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 7, but not 8. 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.

Recurrence (for n>=9): (n-8)*(n^7 - 36*n^6 + 706*n^5 - 13080*n^4 + 177169*n^3 - 1264884*n^2 + 3776364*n - 9605520)*a(n) = (n^8 - 44*n^7 + 802*n^6 - 12104*n^5 + 206449*n^4 - 2516636*n^3 + 16735788*n^2 - 68051376*n + 170709120)*a(n-1) + (n-9)*(n-7)*(n^7 - 29*n^6 + 511*n^5 - 10055*n^4 + 131224*n^3 - 805316*n^2 + 1729104*n - 6929280)*a(n-2). - Vaclav Kotesovec, Mar 11 2014

a(n) ~ sqrt(2)/11520 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1+7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 11 2014

Example

a(7) = 1: [1,2,3,4,5,6,7].
a(8) = 1: [1,2,3,4,5,6,7,1].
a(9) = 3: [1,2,3,4,5,6,7,1,1], [1,2,3,4,5,6,7,1,2], [1,2,3,4,5,6,7,1,8].
a(10) = 9: [1,2,3,4,5,6,7,1,1,1], [1,2,3,4,5,6,7,1,1,2], [1,2,3,4,5,6,7,1,1,8], [1,2,3,4,5,6,7,1,2,1], [1,2,3,4,5,6,7,1,2,3], [1,2,3,4,5,6,7,1,2,8], [1,2,3,4,5,6,7,1,8,1], [1,2,3,4,5,6,7,1,8,2], [1,2,3,4,5,6,7,1,8,9].

Maple

b:= proc(n) option remember; `if`(n<4, [1, 1, 3, 9][n+1],
      ((41*n^2 +82925*n -562776)*b(n-1)
       +(174*n^3 +63287*n^2 -447840*n +352440) *b(n-2)
       +(133*n^3 -81472*n^2 +363510*n -267096) *b(n-3)
       -(n-4)*(30661*n^2 -93002*n -90720) *b(n-4))/
       (174*n^2+31449*n-246768))
    end:
a:=n-> `if`(n<7, 0, b(n-7)):
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 == 7, 1, b[n - 8, {2, 1, 1, 1, 1, 1, 1}]]; a[n_ /; n < 7] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Crossrefs

Column k=7 of A238802.

Sequence in context: A231291 A239116 A239117 * A239119 A238803 A148940

Adjacent sequences: A239115 A239116 A239117 * A239119 A239120 A239121

Keyword

nonn,easy

Author

Joerg Arndt and Alois P. Heinz, Mar 10 2014

Status

approved