A339095 Triangle read by rows: T(n,k) is the number of partitions of n with product of parts equal to k, 1 <= k <= A000792(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 0, 3, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1

Offset: 1

Author

Abhimanyu Kumar, Nov 23 2020

Keywords

• nonn
• tabf

Comments

The product of the parts of a partition is called its norm.

Examples

			For n=6 the partitions and their counts for each norm are given in the table below.
  Relevant partition(s)  | Norm | Count
  1+1+1+1+1+1+1          | 1    | 1
  2+1+1+1+1              | 2    | 1
  3+1+1+1                | 3    | 1
  4+1+1, 2+2+1+1         | 4    | 2
  5+1                    | 5    | 1
  6, 3+2+1               | 6    | 2
  4+2, 2+2+2             | 8    | 2
  3+3                    | 9    | 1
The number of partitions of 6 with norm value 4 are 2, expressed as T(6,4)=2. Similarly, T(6,7)=0 because there is no partition of 6 with norm 7.
So the 6th row is 1, 1, 1, 2, 1, 2, 0, 2, 1.
First few rows of the array are:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1, 2;
  1, 1, 1, 2, 1, 1;
  1, 1, 1, 2, 1, 2, 0, 2, 1;
  1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2;
  1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1;
  ...
		

References

• Abhimanyu Kumar and Meenakshi Rana, On the treatment of partitions as factorization and further analysis, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).

Links

• Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019.

Crossrefs

Cf. A000041 (row sums), A000792 (row lengths), A001055, A118851, A212721.

Programs

PARI

row(n) = {my(list = List()); forpart(p=n, listput(list, vecprod(Vec(p)));); my(vlist = Vec(list)); my(v = vector(vecmax(vlist))); for (i=1, #vlist, v[vlist[i]]++); v;} \\ Michel Marcus, Nov 26 2020

Formula

Let the number of partitions of n having the norm value k refer to the norm counting function T(n,k). The following properties hold true:

Max_{n=1..oo} T(n,k) = A001055(k).

Sum_{k=1..A000792(n)} T(n,k) = A000041(n).

Sum_{n>=1} T(n+1,k) - T(n,k) = A001055(k) - 1.

G.f.: Sum_{n>=0} Sum_{k>=1} ((q^n)/(k^s))*T(n,k) = Product_{m>=1}(1-((q^m)/(m^s)))^(-1).

n*Sum_{k=1..A000792(n)} T(n,k) = Sum_{m=1..n} (Sum_{k=1..A000792(n-m)} T(n-m,i)*k^(-s))*(Sum_{d|m} (d/m)^(d*s-1))

Extensions

More terms from Michel Marcus, Nov 26 2020