A238979 - OEIS

%I #17 Feb 06 2017 18:00:15

%S 0,0,0,0,1,1,3,9,29,98,350,1302,5062,20380,85108,366444,1627836,

%T 7430360,34855016,167546408,825185448,4155400720,21388745008,

%U 112355110672,602103194448,3287743832352,18285157048544,103480813034336,595671084096608,3485006638408128

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

%C The fixed points are in the first 4 positions.

%C 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.

%C 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

%H Joerg Arndt and Alois P. Heinz, <a href="/A238979/b238979.txt">Table of n, a(n) for n = 0..800</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>

%F See Maple program.

%F 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

%F 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

%e a(4) = 1: [1,2,3,4].

%e a(5) = 1: [1,2,3,4,1].

%e a(6) = 3: [1,2,3,4,1,1], [1,2,3,4,1,2], [1,2,3,4,1,5].

%e 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].

%p b:= proc(n) option remember; `if`(n<3, [1$2, 3][n+1],

%p       ((11-n)*b(n-1) +(n^3+4*n^2-15)*b(n-2)

%p        +(n-1)*(n-3)*(n+7)*b(n-3))/((n-1)*(n+1)))

%p     end:

%p a:= n-> `if`(n<4, 0, b(n-4)):

%p seq(a(n), n=0..40);

%t 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 *)

%Y Column k=4 of A238802.

%K nonn,easy

%O 0,7

%A _Joerg Arndt_ and _Alois P. Heinz_, Mar 07 2014