A219311 Number T(n,k) of standard Young tableaux for partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 9, 0, 1, 14, 16, 0, 1, 34, 35, 0, 1, 55, 134, 0, 1, 125, 435, 0, 1, 209, 1213, 768, 0, 1, 461, 3454, 2310, 0, 1, 791, 10484, 11407, 0, 1, 1715, 28249, 44187, 0, 1, 3002, 80302, 200044, 0, 1, 6434, 231895, 680160, 292864
Offset: 0
Examples
A(4,2) = 3: +---------+ +---------+ +---------+ | 1 2 3 | | 1 2 4 | | 1 3 4 | | 4 .-----+ | 3 .-----+ | 2 .-----+ +---+ +---+ +---+ Triangle T(n,k) begins: 1; 0, 1; 0, 1; 0, 1, 2; 0, 1, 3; 0, 1, 9; 0, 1, 14, 16; 0, 1, 34, 35; 0, 1, 55, 134; 0, 1, 125, 435; 0, 1, 209, 1213, 768; 0, 1, 461, 3454, 2310; 0, 1, 791, 10484, 11407; ...
Links
- Alois P. Heinz, Rows n = 0..100, flattened
- Wikipedia, Young tableau
Crossrefs
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:= proc(n, i, k, l) `if`(n=0, h(l), `if`(n>k*(i-(k-1)/2), 0, g(n, i-1, min(k, i-1), l)+`if`(i>n, 0, g(n-i, i-1, k-1, [l[], i])))) end: A:= proc(n, k) option remember; `if`(k<0, 0, g(n, n, k, [])) end: T:= (n, k)-> A(n, k) -A(n, k-1): seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20); -
Mathematica
h[l_] := With[{n = Length[l]}, Sum[i, {i, 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[n_, i_, k_, l_] := If[n == 0, h[l], If[n > k*(i-(k-1)/2), 0, g[n, i-1, Min[k, i-1], l] + If[i > n, 0, g[n-i, i-1, k-1, Append[l, i]]]]]; a[n_, k_] := a[n, k] = If[k < 0, 0, g[n, n, k, {}]]; t[n_, k_] := a[n, k] - a[n, k-1]; Table[Table[t[n, k], {k, 0, Floor[(Sqrt[1+8*n]-1)/2]}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)
Comments