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, 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, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 0, 3, 0, 1, 1, 4, 0, 2, 0, 2, 1, 0, 0, 4, 1, 0, 1, 0, 0, 1, 0, 3, 0, 0, 0, 2

Offset

0,11

Comments

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

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

Alois P. Heinz, Rows n = 0..27, flattened

Abhimanyu Kumar and Meenakshi Rana, Statistical inequalities on partition norms, Ramanujan J. 69 (2026), 79.

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

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)).

For i=0..5: Sum_{k=1..A000792(n)} k^i * T(n,k) give: A000041, A006906, A077335, A265837, A265838, A265839. - Alois P. Heinz, Mar 25 2026

Example

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, 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;
  ...

Maple

b:= proc(n, i, p) option remember; `if`(n=0 or i=1, x^p,
      b(n, i-1, p)+b(n-i, min(n-i, i), p*i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 1)):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 25 2026

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, x^p, b[n, i-1, p] + b[n-i, Min[n-i, i], p*i]];
T[n_] := With[{p = b[n, n, 1]}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 02 2026, after Alois P. Heinz *)

Prog

(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

Crossrefs

Cf. A000041 (row sums), A000792 (row lengths), A001055, A006906, A077335, A118851, A212721, 265837, A265838, A265839, A292193.

T(n,n+1) gives A380218.

Alternating row sums give A394589.

Sequence in context: A276438 A338274 A210960 * A193509 A331284 A331591

Adjacent sequences: A339092 A339093 A339094 * A339096 A339097 A339098

Keyword

nonn,look,tabf

Author

Abhimanyu Kumar, Nov 23 2020

Extensions

More terms from Michel Marcus, Nov 26 2020

T(0,1)=1 prepended by Alois P. Heinz, Mar 25 2026

Status

approved