A245001 Number of standard Young tableaux with n cells and 3 as last value in the first row.

2, 3, 5, 10, 19, 41, 86, 197, 449, 1087, 2650, 6722, 17227, 45267, 120069, 323442, 877777, 2405399, 6628760, 18384040, 51204735, 143252991, 402115301, 1132464571, 3197928097, 9053803101, 25689876776, 73047889402, 208100836969, 593897902349, 1697686011406

Offset

3,1

Comments

Also the number of ballot sequences of length n where 3 is the position of the last occurrence of the minimal value.

Links

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 3..1000

Wikipedia, Young tableau

Formula

Recurrence: see Maple program.

a(n) ~ 3^(n-3/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 11 2014

Example

a(4) = 3:
[1 3]  [1 3]  [1 2 3]
[2]    [2 4]  [4]
[4]

Maple

a:= proc(n) option remember; `if`(n<5, [0$3, 2, 3][n+1],
     ((n-2)*(30*n^4-505*n^3+3108*n^2-8147*n+7338)*a(n-1)
     +(18703*n^3-76648*n^2+154520*n-122616-2240*n^4+105*n^5)*a(n-2)
     -2*(n-5)*(60*n^4-965*n^3+5766*n^2-15082*n+14364)*a(n-3)
     -12*(n-5)*(n-6)*(15*n^3-185*n^2+744*n-994)*a(n-4)) /
     ((n-1)*(n-2)*(15*n^3-230*n^2+1159*n-1938)))
    end:
seq(a(n), n=3..40);

Mathematica

b[n_, l_List] := b[n, l] = If[n == 0, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + Function[{p}, p+(x^(1 + Total[l])-1)*Coefficient[p, x, 0]][b[n-1, Append[l, 1]]]]; a[n_] := Coefficient[ b[n, {}], x, 3]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 3, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple code in A238794 *)

Crossrefs

Column k=3 of A238794.

Sequence in context: A293328 A064236 A339585 * A365858 A007569 A054317

Adjacent sequences: A244998 A244999 A245000 * A245002 A245003 A245004

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Jul 09 2014

Status

approved