A238728 Number of standard Young tableaux with n cells where the largest value n is contained in the last row.
1, 1, 2, 3, 7, 14, 41, 107, 337, 1066, 3691, 12962, 49061, 188894, 766845, 3182844, 13758383, 60858842, 278312475, 1301323108, 6258671365, 30742575588, 154785692507, 794888735945, 4173162573277, 22318859784416, 121767607626621, 676010926754742
Offset: 0
Keywords
Examples
a(4) = 7 counts the following tableaux: [1] [1 2] [1 3] [1 2 3] [1 2] [1 3] [1 2 3 4] [2] [3] [2] [4] [3 4] [2 4] [3] [4] [4] [4] corresponding to the following ballot sequences: [1,2,3,4], [1,1,2,3], [1,2,1,3], [1,1,1,2], [1,1,2,2], [1,2,1,2], [1,1,1,1].
Links
- Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..60
- Wikipedia, Young tableau
Programs
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Maple
h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+ add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end: g:= l->`if`(l=[], 1, h(subsop(-1=`if`(l[-1]=1, [][], l[-1]-1), l))): b:= proc(n, i, l) `if`(n=0 or i=1, g([l[], 1$n]), add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)) end: a:= n-> b(n, n, []): seq(a(n), n=0..28); -
Mathematica
h[l_List] := With[{n=Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[l_List] := If[l == {}, 1, h[If[Last[l] == 1, Most[l], Append[Most[l], Last[l]-1]]]]; b[n_, i_, l_List] := If[n == 0 || i == 1, g[Join[l, Array[1&, n]]], Sum[b[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]; a[n_] := b[n, n, {}]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Feb 12 2015, after Maple *)
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