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%I #5 Apr 11 2019 21:00:09
%S 0,0,2,0,0,2,4,2,2,4,8,10,12,10,14,20,28,36,44,46,56,66,86,108,136,
%T 160,190,214,252,298,364,434,524,620,728,834,966,1112,1306,1522,1788,
%U 2088,2448,2822,3256,3720,4264,4876,5610,6434,7420
%N Number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 2.
%C The Heinz numbers of these partitions are given by A325186.
%C The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k from the lower-right boundary. For example, the partition (6,5,5,3) has diagram
%C o o o o o o
%C o o o o o
%C o o o o o
%C o o o
%C with origin-to-boundary graph-distances
%C 4 4 4 3 2 1
%C 3 3 3 2 1
%C 2 2 2 1 1
%C 1 1 1
%C giving the origin-to-boundary partition (7,5,4,3).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.
%e The a(2) = 1 through a(11) = 10 partitions:
%e (2) (32) (33) (52) (62) (72) (82) (92)
%e (11) (221) (42) (22111) (221111) (432) (433) (443)
%e (222) (3321) (442) (533)
%e (2211) (2211111) (532) (542)
%e (3322) (632)
%e (3331) (3332)
%e (33211) (33221)
%e (22111111) (33311)
%e (332111)
%e (221111111)
%t ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];
%t corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];
%t Table[Length[Select[IntegerPartitions[n],Apply[Plus,If[#=={},{},FixedPointList[corpos,ptnmat[#]][[-3]]],{0,1}]==2&]],{n,30}]
%Y Cf. A188674, A325165, A325169, A325183, A325184, A325186, A325187, A325190, A325199.
%K nonn
%O 0,3
%A _Gus Wiseman_, Apr 11 2019
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