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A238979
Number of ballot sequences of length n with exactly 4 fixed points.
2
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
KEYWORD
nonn,easy
AUTHOR
Joerg Arndt and Alois P. Heinz, Mar 07 2014
STATUS
approved