A322480 Irregular triangular array read by rows: T(n,k), n>=1, is the number of ordered factorizations corresponding to each unordered factorization, indexed by k.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 2, 2, 2, 3, 6, 4, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 2, 2, 6, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 3, 1, 6, 3, 6, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 6, 4, 1, 1, 2, 2, 2, 6, 1, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 1, 2, 2, 2, 3, 2, 6, 6, 4, 12, 5

Offset: 1

Author

Thomas Anton, Dec 09 2018

Keywords

• nonn
• tabf

Comments

The method of indexing the unordered factorizations of n in this array is as follows: take all unordered factorizations of n and write them with their factors in nonincreasing order (e.g., 2*4*5*3 becomes 5*4*3*2), and order these reverse-lexicographically (e.g., for 12: 12, 6*2, 4*3, 3*2*2), then assign the index k to the k-th factorization in this ordering.

For a sequence f with Dirichlet inverse f^(-1), f^(-1)(n) is the sum over all multisets M of integers > 1 with product n, of the product of the terms f(m) with indices m in M (counted with multiplicity) multiplied by T(n,k)*(-1)^c/f(1)^(c+1) where c = |M| and T(n,k) corresponds to M.

The multiset of entries in the n-th row is determined by the prime signature of n.

For the p^j-th row with p a prime, the entries give the number of compositions of j corresponding to each partition of j, indexed by k in an analogous manner, given by the j-th row of A048996.

Examples

			  1;
  1;
  1;
  1, 1;
  1;
  1, 2;
  1;
  1, 2, 1;
  1, 1;
  1, 2;
  1;
  1, 2, 2, 3;
  etc.
The 12th row is 1,2,2,3, because 12 can be factored as 12, 6*2, 3*4 or 3*2*2 with respective sets of ordered factorizations {12}, {6*2, 2*6}, {4*3, 3*4} and {3*2*2, 2*3*2, 2*2*3}, with respective cardinalities 1, 2, 2 and 3.
		

Crossrefs

Cf. A048996, A002033 (row sums), A212171, A251683, A001055 (row lengths).