cp's OEIS Frontend

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

%I A265947 #47 Apr 06 2020 15:53:20
%S A265947 1,1,3,6,14,26,55,99,192,340,619,1063,1873,3129,5308,8718,14385,23116,
%T A265947 37346,58949,93294,145131,225623,345833,529976,801675,1211225,1811558,
%U A265947 2703327,3998289,5901849,8641160,12623450,18315370,26503133,38119289,54691750,78028166,111041918,157250528,222105633
%N A265947 Total size of all principal order ideals in the poset of integer partitions of n with the refinement order.
%C A265947 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 A265947 Jon Mark Perry et al., <a href="http://mathoverflow.net/questions/226656/counting-refinements-of-partitions">Counting refinements of partitions</a>, Mathoverflow, 2015.
%e A265947 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 A265947 (Sage)
%o A265947 def A265947(n):
%o A265947     P = Posets.IntegerPartitions(n)
%o A265947     return sum( len(P.order_ideal([p])) for p in P )
%o A265947 (Sage) # Alternative:
%o A265947 def A265947(n):
%o A265947     return Posets.IntegerPartitions(n).relations_number() # _F. Chapoton_, Feb 26 2020
%Y A265947 Cf. A001764, A002846, A213242, A213385, A213427, A063834.
%K A265947 nonn
%O A265947 0,3
%A A265947 _Max Alekseyev_, Dec 23 2015