A380959 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of k with product n.

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, Feb 10 2025

Keywords

• nonn
• tabl

Comments

Counts finite multisets of positive integers by product and sum.

Examples

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
       ---------------------------------------------------
   n=1: 1   1   1   1   1   1   1   1   1   1   1   1   1
   n=2: 0   0   1   1   1   1   1   1   1   1   1   1   1
   n=3: 0   0   0   1   1   1   1   1   1   1   1   1   1
   n=4: 0   0   0   0   2   2   2   2   2   2   2   2   2
   n=5: 0   0   0   0   0   1   1   1   1   1   1   1   1
   n=6: 0   0   0   0   0   1   2   2   2   2   2   2   2
   n=7: 0   0   0   0   0   0   0   1   1   1   1   1   1
   n=8: 0   0   0   0   0   0   2   2   3   3   3   3   3
   n=9: 0   0   0   0   0   0   1   1   1   2   2   2   2
  n=10: 0   0   0   0   0   0   0   1   1   1   2   2   2
  n=11: 0   0   0   0   0   0   0   0   0   0   0   1   1
  n=12: 0   0   0   0   0   0   0   2   3   3   3   3   4
The A(12,9) = 3 partitions are: (6,2,1), (4,3,1,1), (3,2,2,1,1).
The A(9,12) = 2 partitions are: (9,1,1,1), (3,3,1,1,1,1,1,1).
		

Crossrefs

Column sums are A000041 = partitions of n, strict A000009, no ones A002865.

Diagonal A(n,n) is A001055(n) = factorizations of n, strict A045778.

Row n converges to A001055(n).

Lower triangle is A319000.

Transpose of A379666.

Antidiagonal sums are A379667, without ones A379669 (zeros A379670), strict A379672.

A316439 counts factorizations by length, partitions A008284.

A326622 counts factorizations with integer mean, strict A328966.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A379733

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029

- less or equal: A319005, ranks A379721, see A025147

- different: A379736, ranks A379722, see A111133

Cf. A003963, A028422, A069016, A319916, A319057, A325036, A325041, A325042, A326152.

Cf. A318950, A379668, A379671, A379678.

Programs

Mathematica

nn=12;
tt=Table[Length[Select[IntegerPartitions[k],Times@@#==n&]],{n,1,nn},{k,0,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

Formula

A(n,k) = A379666(k,n).