login
A239118
Number of ballot sequences of length n with exactly 7 fixed points.
2
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
KEYWORD
nonn,easy
AUTHOR
Joerg Arndt and Alois P. Heinz, Mar 10 2014
STATUS
approved