A379678 Array read by antidiagonals downward where A(n,k) is the number of finite sets of positive integers > 1 with sum n and 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, 1, 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

Comments

Counts finite sets of positive integers > 1 by sum and product. Compare to the triangle A379737.

Links

Table of n, a(n) for n=1..78.

Formula

For k >= n we have A(n,k) = A379737(k,n).

Example

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   1   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   1   0   0   0   0
   n=7:  0   0   0   0   0   0   1   0   0   1   0   1
   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(17,72) = 2 sets are: {2,3,12}, {8,9}.
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 1 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 1 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 0 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 sets:
  n=6: {2,4}
  n=7: {7}
so the 14th antidiagonal is: (0,0,0,0,0,0,1,1,0,0,0,0,0,0).
Antidiagonal n+k=89 counts the following sets:
  n=17: {2,3,12}, {8,9}
  n=19: {5,14}
  n=21: {4,17}
  n=31: {2,29}
so the 89th antidiagonal is: (...,0,0,0,2,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,...).

Mathematica

nn=12;
tt=Table[Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&UnsameQ@@#&&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 *)

Crossrefs

Row sums are A025147 = strict partitions into parts > 1, non-strict A002865.

Column sums are A045778 = strict factorizations, non-strict A001055.

The non-strict version allowing ones is A379666, antidiagonal sums A379667.

The non-strict version is A379668, antidiagonal sums A379669 (zeros A379670).

The version allowing ones is A379671, antidiagonal sums A379672.

Antidiagonal sums are A379679 (zeros A379680).

Take transpose and remove upper half (all zeros) to get A379737.

A000041 counts integer partitions, strict A000009.

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 A326156, A326172

- 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

- different: A379736, ranks A379722, see A111133

Cf. A028422, A069016, A318950, A319057, A325036, A325041, A325042, A326152, A326178, A379720, A379733.

Sequence in context: A359160 A373978 A326499 * A104124 A347246 A052434

Adjacent sequences: A379675 A379676 A379677 * A379679 A379680 A379681

Keyword

nonn,tabl

Author

Gus Wiseman, Jan 01 2025

Status

approved