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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).

%I A339095 #61 Nov 27 2020 04:00:07
%S A339095 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,
%T A339095 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,
%U A339095 1,1,3,0,2,0,2,0,0,0,2,0,0,1
%N 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).
%C A339095 The product of the parts of a partition is called its norm.
%D A339095 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).
%H A339095 Andrew V. Sills and Robert Schneider, <a href="https://arxiv.org/abs/1904.08004">The product of parts or "norm" of a partition</a>, arXiv:1904.08004 [math.NT], 2019.
%F A339095 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:
%F A339095 Max_{n=1..oo} T(n,k) = A001055(k).
%F A339095 Sum_{k=1..A000792(n)} T(n,k) = A000041(n).
%F A339095 Sum_{n>=1} T(n+1,k) - T(n,k) = A001055(k) - 1.
%F A339095 G.f.: Sum_{n>=0} Sum_{k>=1} ((q^n)/(k^s))*T(n,k) = Product_{m>=1}(1-((q^m)/(m^s)))^(-1).
%F A339095 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))
%e A339095 For n=6 the partitions and their counts for each norm are given in the table below.
%e A339095   Relevant partition(s)  | Norm | Count
%e A339095   1+1+1+1+1+1+1          | 1    | 1
%e A339095   2+1+1+1+1              | 2    | 1
%e A339095   3+1+1+1                | 3    | 1
%e A339095   4+1+1, 2+2+1+1         | 4    | 2
%e A339095   5+1                    | 5    | 1
%e A339095   6, 3+2+1               | 6    | 2
%e A339095   4+2, 2+2+2             | 8    | 2
%e A339095   3+3                    | 9    | 1
%e A339095 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.
%e A339095 So the 6th row is 1, 1, 1, 2, 1, 2, 0, 2, 1.
%e A339095 First few rows of the array are:
%e A339095   1;
%e A339095   1, 1;
%e A339095   1, 1, 1;
%e A339095   1, 1, 1, 2;
%e A339095   1, 1, 1, 2, 1, 1;
%e A339095   1, 1, 1, 2, 1, 2, 0, 2, 1;
%e A339095   1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2;
%e A339095   1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1;
%e A339095   ...
%o A339095 (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
%Y A339095 Cf. A000041 (row sums), A000792 (row lengths), A001055, A118851, A212721.
%K A339095 nonn,tabf
%O A339095 1,10
%A A339095 _Abhimanyu Kumar_, Nov 23 2020
%E A339095 More terms from _Michel Marcus_, Nov 26 2020