OFFSET
1,2
COMMENTS
Given a partition pi = (p1, p2, p3, ...) we call the associated multinomial coefficient (p1+p2+ ...)! / (p1!*p2!*p3! ...) the 'partition coefficient' of pi and denote it by <pi>. We say 'k is represented by pi' if k = <pi>.
A partition is a prime partition if all parts are prime.
LINKS
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, arXiv:math/0509470 [math.CO], 2005.
Eric Weisstein's World of Mathematics, Prime Partition
EXAMPLE
(2*n)!/2^n (for n >= 1) is a subsequence because [2,2,...,2] (n times '2') is a prime partition. Similarly A327411(n) is a subsequence because [3,2,2,...,2] (n times '2') is a prime partition. (3*n)!/(6^n) and A327412 are subsequences for the same reason.
The representations are not unique. 1 is the represented by all partitions of the form [p], p prime. For example 210 is represented by [3, 2, 2] and by [19, 2]. The list below shows the partitions with the smallest sum.
1 <- [2],
6 <- [2, 2],
10 <- [3, 2],
20 <- [3, 3],
21 <- [5, 2],
36 <- [7, 2],
56 <- [5, 3],
78 <- [11, 2],
90 <- [2, 2, 2],
105 <- [13, 2],
120 <- [7, 3],
171 <- [17, 2],
210 <- [3, 2, 2],
252 <- [5, 5],
300 <- [23, 2].
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Sep 07 2019
STATUS
approved