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A239117
Number of ballot sequences of length n with exactly 6 fixed points.
2
0, 0, 0, 0, 0, 0, 1, 1, 3, 9, 29, 99, 357, 1350, 5334, 21912, 93352, 410988, 1866492, 8720924, 41866020, 206085480, 1039120104, 5358418224, 28235017104, 151838491408, 832730775888, 4652886489840, 26470731088016, 153207256585824, 901628675631456, 5391563546317056
OFFSET
0,9
COMMENTS
The fixed points are in the first 6 positions.
Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 6, but not 7. 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>=8): (n-7)*(n^6 - 28*n^5 + 350*n^4 - 3850*n^3 + 37569*n^2 - 201082*n + 556800)*a(n) = (n^7 - 35*n^6 + 441*n^5 - 3220*n^4 + 31444*n^3 - 314265*n^2 + 1921954*n - 5066880)*a(n-1) + (n-8)*(n-6)*(n^6 - 22*n^5 + 225*n^4 - 2710*n^3 + 27854*n^2 - 136228*n + 389760)*a(n-2). - Vaclav Kotesovec, Mar 11 2014
a(n) ~ sqrt(2)/1680 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1+7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 11 2014
EXAMPLE
a(6) = 1: [1,2,3,4,5,6].
a(7) = 1: [1,2,3,4,5,6,1].
a(8) = 3: [1,2,3,4,5,6,1,1], [1,2,3,4,5,6,1,2], [1,2,3,4,5,6,1,7].
a(9) = 9: [1,2,3,4,5,6,1,1,1], [1,2,3,4,5,6,1,1,2], [1,2,3,4,5,6,1,1,7], [1,2,3,4,5,6,1,2,1], [1,2,3,4,5,6,1,2,3], [1,2,3,4,5,6,1,2,7], [1,2,3,4,5,6,1,7,1], [1,2,3,4,5,6,1,7,2], [1,2,3,4,5,6,1,7,8].
MAPLE
b:= proc(n) option remember; `if`(n<3, [1, 1, 3][n+1],
((114*n^3 -68193*n^2 +266129*n -764878)*b(n-1)
+(2513*n^4 +25106*n^3 +330108*n^2 -382379*n +208440)*b(n-2)
+(n-3)*(2399*n^3 +95128*n^2 +269793*n +65880)*b(n-3))/
(2513*n^3+10142*n^2+201063*n-630958))
end:
a:=n-> `if`(n<6, 0, b(n-6)):
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 == 6, 1, b[n - 7, {2, 1, 1, 1, 1, 1}]]; a[n_ /; n < 6] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)
CROSSREFS
Column k=6 of A238802.
Sequence in context: A074526 A231291 A239116 * A239118 A239119 A238803
KEYWORD
nonn,easy,changed
AUTHOR
Joerg Arndt and Alois P. Heinz, Mar 10 2014
STATUS
approved