

A265947


Total size of all principal order ideals in the poset of integer partitions of n with the refinement order.


30



1, 1, 3, 6, 14, 26, 55, 99, 192, 340, 619, 1063, 1873, 3129, 5308, 8718, 14385, 23116, 37346, 58949, 93294, 145131, 225623, 345833, 529976, 801675, 1211225, 1811558, 2703327, 3998289, 5901849, 8641160, 12623450, 18315370, 26503133, 38119289, 54691750, 78028166, 111041918, 157250528, 222105633
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OFFSET

0,3


COMMENTS

a(n) is the number of refinementordered 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

Table of n, a(n) for n=0..40.
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

Cf. A001764, A002846, A213242, A213385, A213427, A063834.
Sequence in context: A324703 A120940 A049940 * A323450 A051749 A278788
Adjacent sequences: A265944 A265945 A265946 * A265948 A265949 A265950


KEYWORD

nonn


AUTHOR

Max Alekseyev, Dec 23 2015


STATUS

approved



