OFFSET
0,3
COMMENTS
a(n) is the number of refinement-ordered pairs of integer partitions of n. Every such pair (x,y) is a multiset union x and a multiset of sums y of some weakly ordered sequence of integer partitions, so this sequence is dominated by A063834 (twice partitioned numbers). - Gus Wiseman, May 01 2016
LINKS
Jon Mark Perry et al., Counting refinements of partitions, Mathoverflow, 2015.
EXAMPLE
a(4) = 14 ordered pairs of partitions: {(4,4), (4,22), (4,31), (4,211), (4,1111), (22,22), (22,211), (22,1111), (31,31), (31,211), (31,1111), (211,211), (211,1111), (1111,1111)}.
PROG
(Sage)
def A265947(n):
P = Posets.IntegerPartitions(n)
return sum( len(P.order_ideal([p])) for p in P )
(Sage) # Alternative:
def A265947(n):
return Posets.IntegerPartitions(n).relations_number() # F. Chapoton, Feb 26 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Max Alekseyev, Dec 23 2015
STATUS
approved