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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified October 1 04:06 EDT 2020. Contains 337441 sequences. (Running on oeis4.)