A379668 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n into parts > 1 with product k.

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

Offset: 1

Author

Gus Wiseman, Dec 31 2024

Keywords

• nonn
• tabl

Comments

This table counts finite multisets of positive integers > 1 by sum and product. Compare to the triangle A318950.

Examples

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  0   0   0   0   0   0   0   0   0   0   0   0
   n=2:  0   1   0   0   0   0   0   0   0   0   0   0
   n=3:  0   0   1   0   0   0   0   0   0   0   0   0
   n=4:  0   0   0   2   0   0   0   0   0   0   0   0
   n=5:  0   0   0   0   1   1   0   0   0   0   0   0
   n=6:  0   0   0   0   0   1   0   2   1   0   0   0
   n=7:  0   0   0   0   0   0   1   0   0   1   0   2
   n=8:  0   0   0   0   0   0   0   1   0   0   0   1
   n=9:  0   0   0   0   0   0   0   0   1   0   0   0
  n=10:  0   0   0   0   0   0   0   0   0   1   0   0
  n=11:  0   0   0   0   0   0   0   0   0   0   1   0
  n=12:  0   0   0   0   0   0   0   0   0   0   0   1
For example, the A(11,48) = 3 partitions are: (4,4,3), (4,3,2,2), (3,2,2,2,2).
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 0
   n+k=3: 0 0 0
   n+k=4: 0 0 1 0
   n+k=5: 0 0 0 0 0
   n+k=6: 0 0 0 1 0 0
   n+k=7: 0 0 0 0 0 0 0
   n+k=8: 0 0 0 0 2 0 0 0
   n+k=9: 0 0 0 0 0 0 0 0 0
  n+k=10: 0 0 0 0 0 1 0 0 0 0
  n+k=11: 0 0 0 0 0 1 0 0 0 0 0
  n+k=12: 0 0 0 0 0 0 1 0 0 0 0 0
  n+k=13: 0 0 0 0 0 0 0 0 0 0 0 0 0
  n+k=14: 0 0 0 0 0 0 2 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
  n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
For example, antidiagonal n+k=14 counts the following partitions:
  n=6: (42), (222)
  n=7: (7)
so the 14th antidiagonal is: (0,0,0,0,0,0,2,1,0,0,0,0,0,0,0).
		

Crossrefs

Column sums are A001055 = factorizations, strict A045778.

Row sums are A002865 = partitions into parts > 1.

Take transpose and remove upper half (all zeros) to get A318950.

Allowing one gives A379666, antidiagonal sums A379667.

Antidiagonal sums are A379669, zeros A379670.

The strict case allowing ones is A379671, antidiagonal sums A379672.

The strict case is A379678, antidiagonal sums A379679 (zeros A379680).

A000041 counts integer partitions, strict A000009.

A316439 counts factorizations by length, A008284 partitions.

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, A319000, A319057, A319916, A325036, A325041, A325042, A326152, A326178, A379720.

Programs

Mathematica

nn=15;
tt=Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

Formula

For n <= k we have A(n,k) = A318950(k,n).