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

0,3

Comments

Row sums give A000041.

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

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.

Formula

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

Example

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;
...

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 *)

Crossrefs

Cf. A000005, A000041, A130519, A271370.

Sequence in context: A193262 A120967 A116687 * A236293 A056044 A236097

Adjacent sequences: A264030 A264031 A264032 * A264034 A264035 A264036

Keyword

nonn,tabf

Author

Christian Stump, Nov 01 2015

Status

approved