A264033 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A130519(n+1)) is the number of integer partitions of n having k pairs of different size.

1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 1, 4, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 1, 4, 3, 3, 3, 2, 3, 2, 1, 1, 3, 4, 3, 5, 2, 5, 2, 2, 1, 2, 1, 4, 4, 4, 7, 3, 4, 2, 4, 5, 1, 0, 2, 2, 2, 5, 5, 8, 2, 9, 4, 4, 3, 4, 1, 4, 1, 1, 2, 1, 6, 5, 4, 9, 4, 9, 4, 6, 5, 7, 2, 4, 3, 1, 2, 2, 2, 1, 1

Offset: 0

Author

Christian Stump, Nov 01 2015

Keywords

• nonn
• tabf

Comments

Row sums give A000041.

T(n,0) gives A000005(n) for n>0. - Alois P. Heinz, Nov 01 2015

Examples

			Triangle begins:
1;
1;
2;
2,1;
3,1,1;
2,2,2,1;
4,2,1,2,2;
2,3,3,3,1,2,1;
4,3,3,3,2,3,2,1,1;
3,4,3,5,2,5,2,2,1,2,1;
4,4,4,7,3,4,2,4,5,1,0,2,2;
2,5,5,8,2,9,4,4,3,4,1,4,1,1,2,1;
6,5,4,9,4,9,4,6,5,7,2,4,3,1,2,2,2,1,1;
...
		

References

• Richard Stanley, Enumerative combinatorics. Vol. 2 MathSciNet:1676282, page 375.

Links

• Alois P. Heinz, Rows n = 0..80, flattened
• FindStat - Combinatorial Statistic Finder, Degree of the polynomial counting the number of semistandard Young-tableaux of shape k*lambda.

Crossrefs

Cf. A000005, A000041, A130519, A271370.

Programs

Maple

b:= proc(n, i, p, t) option remember; expand(
      `if`(n=0, x^t, `if`(i<1, 0, add(
       b(n-i*j, i-1, p+j, t+j*p), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Nov 01 2015

Mathematica

b[n_, i_, p_, t_] := b[n, i, p, t] = Expand[If[n==0, x^t, If[i<1, 0, Sum[b[n-i*j, i-1, p+j, t+j*p], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)

Formula

Sum_{k>0} k * T(n,k) = A271370(n). - Alois P. Heinz, Apr 05 2016