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

Author

Max Alekseyev, Dec 23 2015

Keywords

• nonn

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

Examples

			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)}.
		

Links

• Jon Mark Perry et al., Counting refinements of partitions, Mathoverflow, 2015.

Crossrefs

Cf. A001764, A002846, A213242, A213385, A213427, A063834.

Programs

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