A265947 Total size of all principal order ideals in the poset of integer partitions of n with the refinement order.

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

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

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 * A358831 A323450 A051749

Adjacent sequences: A265944 A265945 A265946 * A265948 A265949 A265950

Keyword

nonn

Author

Max Alekseyev, Dec 23 2015

Status

approved