A219273 -id:A219273 - cp's OEIS frontend

A219272 Number A(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part <= k; triangle A(n,k), k>=0, 0<=n<=k*(k+1)/2, read by columns.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 5, 16, 1, 1, 1, 3, 4, 9, 25, 49, 70, 168, 768, 1, 1, 1, 3, 4, 10, 30, 63, 162, 372, 1506, 3300, 7887, 15015, 48048, 292864, 1, 1, 1, 3, 4, 10, 31, 69, 182, 525, 1911, 5115, 17347, 43758, 149721, 626769, 1946516, 4934930

Offset: 0

Alois P. Heinz, Nov 17 2012

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

			A(3,2) = 2:
+------+  +------+
| 1  2 |  | 1  3 |
| 3 .--+  | 2 .--+
+---+     +---+
A(3,3) = 3:
+------+  +------+  +---------+
| 1  2 |  | 1  3 |  | 1  2  3 |
| 3 .--+  | 2 .--+  +---------+
+---+     +---+
Triangle A(n,k) begins:
1,  1,  1,  1,   1,    1,    1,    1,    1, ...
.   1,  1,  1,   1,    1,    1,    1,    1, ...
.       1,  1,   1,    1,    1,    1,    1, ...
.       2,  3,   3,    3,    3,    3,    3, ...
.           3,   4,    4,    4,    4,    4, ...
.           5,   9,   10,   10,   10,   10, ...
.          16,  25,   30,   31,   31,   31, ...
.               49,   63,   69,   70,   70, ...
.               70,  162,  182,  189,  190, ...

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

Column heights are A000124.
Column sums give: A219273.
Diagonal gives: A218293.
Leftmost nonzero elements give A219339.
Column of leftmost nonzero element is A002024(n) for n>0.
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:
A:= (n, k)-> g(n, k, []):
seq(seq(A(n, k), n=0..k*(k+1)/2), k=0..7);

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}]];
g[n_, i_, l_] := g[n, i, l] = With[{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]]]]]];
A[n_, k_] := g[n, k, {}];
Table[Table[A[n, k], {n, 0, k*(k+1)/2}], {k, 0, 7}] // Flatten (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} A219274(n,i).

A219275 Number of standard Young tableaux for partitions of nonnegative integers into distinct parts with largest part n.

Original entry on oeis.org

1, 1, 3, 25, 1069, 368168, 1299366501, 55208013380403, 32401197537296758130, 297072961835477978342245712, 47538199827835784548062928051198402, 146779873623344672821145371965795071455181183, 9581411392319396646028223743176779937161862866453789852

Offset: 0

Alois P. Heinz, Nov 17 2012

nonn

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

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

Column sums of A219274.

First differences of A219273.

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-> `if`(n=0, 1, b(n-1, [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_] := If[n == 0, 1, b[n - 1, {n}]];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 02 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A219272 Number A(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part <= k; triangle A(n,k), k>=0, 0<=n<=k*(k+1)/2, read by columns.

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