A265947 - OEIS

%I #47 Apr 06 2020 15:53:20

%S 1,1,3,6,14,26,55,99,192,340,619,1063,1873,3129,5308,8718,14385,23116,

%T 37346,58949,93294,145131,225623,345833,529976,801675,1211225,1811558,

%U 2703327,3998289,5901849,8641160,12623450,18315370,26503133,38119289,54691750,78028166,111041918,157250528,222105633

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

%C 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

%H Jon Mark Perry et al., <a href="http://mathoverflow.net/questions/226656/counting-refinements-of-partitions">Counting refinements of partitions</a>, Mathoverflow, 2015.

%e 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)}.

%o (Sage)

%o def A265947(n):

%o     P = Posets.IntegerPartitions(n)

%o     return sum( len(P.order_ideal([p])) for p in P )

%o (Sage) # Alternative:

%o def A265947(n):

%o     return Posets.IntegerPartitions(n).relations_number() # _F. Chapoton_, Feb 26 2020

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

%K nonn

%O 0,3

%A _Max Alekseyev_, Dec 23 2015