A219275 -id:A219275 - cp's OEIS frontend

A219274 Number T(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.

1, 1, 1, 2, 1, 3, 5, 16, 1, 4, 9, 49, 70, 168, 768, 1, 5, 14, 92, 204, 738, 3300, 7887, 15015, 48048, 292864, 1, 6, 20, 153, 405, 1815, 9460, 28743, 101673, 333905, 1946516, 4934930, 14454726, 34918884, 141892608, 1100742656, 1, 7, 27, 235, 715, 3630, 21307

Offset: 0

Alois P. Heinz, Nov 17 2012

T(n,k) is defined for n,k >= 0. T(n,k) = 0 iff n k*(k+1)/2 = A000217(k). The triangle contains only the nonzero terms.

			T(3,2) = 2:
+------+  +------+
| 1  2 |  | 1  3 |
| 3 .--+  | 2 .--+
+---+     +---+
Triangle T(n,k) begins:
1;
.  1;
.     1;
.     2,  1;
.         3,   1;
.         5,   4,   1;
.        16,   9,   5,   1;
.             49,  14,   6,   1;
.             70,  92,  20,   7,  1;
.            168, 204, 153,  27,  8, 1;
.            768, 738, 405, 235, 35, 9, 1;

Alois P. Heinz, Columns k = 0..22, flattened
Wikipedia, Young tableau

Column heights are A000124(k-1) for k>0.
Column sums give: A219275.
Row sums give: A218293.
Diagonal and lower diagonals give: A000012, A000027 (for n>1), A000096(n-1) (for n>2).
Leftmost nonzero elements give A219339.
Column of leftmost nonzero element is A002024(n) for n>0.
Triangle read by rows reversed gives: A219356.
T(A000217(n),n) = A005118(n+1).

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, l) local s; s:=i*(i+1)/2;
      `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
    end:
T:= (n, k)-> `if`(k>n, 0, g(n-k, k-1, [k])):
seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..7);

h[l_] := Module[{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[n_, i_, l_] := Module[{s = i(i + 1)/2}, If[n == s, h[Join[l, Table[i - j, {j, 0, i - 1}]]], If[n > s, 0, g[n, i - 1, l] + If[i > n, 0, g[n - i, i - 1, Append[l, i]]]]]];
T[n_, k_] := If[k > n, 0, g[n - k, k - 1, {k}]];
Table[Table[T[n, k], {n, k, k(k + 1)/2}], {k, 0, 7}] // Flatten (* Jean-François Alcover, Sep 01 2023, after Alois P. Heinz *)

T(n,k) = A219272(n,k) - A219272(n,k-1) for k>0.

A219273 Number of standard Young tableaux for all partitions of nonnegative integers into distinct parts with largest part <= n.

Original entry on oeis.org

1, 2, 5, 30, 1099, 369267, 1299735768, 55209313116171, 32401252746609874301, 297072994236730724952120013, 47538200124908778784793653003318415, 146779873670882872946054150750588724458499598, 9581411392466176519699106616122834087912451590912289450

Offset: 0

Alois P. Heinz, Nov 17 2012

nonn

			a(2) = 5:
+------+  +------+  +------+  +---+  +-+
| 1  2 |  | 1  3 |  | 1  2 |  | 1 |  +-+
| 3 .--+  | 2 .--+  +------+  +---+
+---+     +---+

Alois P. Heinz, Table of n, a(n) for n = 0..22
Wikipedia, Young tableau

Column sums of A219272.

Partial sums of A219275.

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:
b:= (n, l)-> `if`(n<1, h(l), b(n-1, l) +b(n-1, [l[], n])):
a:= n-> b(n, []):
seq(a(n), n=0..12);

h[l_] := 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}]];
b[n_, l_] := If[n < 1, h[l], b[n - 1, l] + b[n - 1, Append[l, n]]];
a[n_] := b[n, {}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 02 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A219274 Number T(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.

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