OFFSET
1,10
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
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_{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).
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, 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;
...
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
KEYWORD
nonn,tabf
AUTHOR
Abhimanyu Kumar, Nov 23 2020
EXTENSIONS
More terms from Michel Marcus, Nov 26 2020
STATUS
approved