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 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified January 17 13:59 EST 2021. Contains 340242 sequences. (Running on oeis4.)