A049400 Partial sums of rows of A047884. Young Tableaux by height.

1, 1, 2, 1, 3, 4, 1, 6, 9, 10, 1, 10, 21, 25, 26, 1, 20, 51, 70, 75, 76, 1, 35, 127, 196, 225, 231, 232, 1, 70, 323, 588, 715, 756, 763, 764, 1, 126, 835, 1764, 2347, 2556, 2611, 2619, 2620, 1, 252, 2188, 5544, 7990, 9096, 9415, 9486, 9495, 9496, 1, 462, 5798, 17424, 27908, 33231, 35135, 35596

Offset: 1

Author

Alford Arnold

Keywords

• easy
• nice
• nonn
• tabl

Examples

			1;
1,  2;
1,  3,   4;
1,  6,   9,  10;
1, 10,  21,  25,  26;
1, 20,  51,  70,  75,  76;
1, 35, 127, 196, 225, 231, 232;
1, 70, 323, 588, 715, 756, 763, 764;
		

Links

• Seiichi Manyama, Rows n = 1..70, flattened (first 44 rows from Alois P. Heinz)
• Index entries for sequences related to Young tableaux.

Crossrefs

Cf. A000085, A007579, A007578, A007580, A049401, A293128.

Programs

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, l) option remember;
      `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]), g(n, i-1, l)+
      `if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
T:= (n, k)-> g(n, k, []):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Apr 16 2012

Mathematica

Accumulate /@ Table[ Plus @@ NumberOfTableaux /@ Reverse /@ Union[ Sort /@ (Compositions[n - m, m] + 1)], {n, 1, 12}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 29 2013, after Mathematica program for A047884 *)