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%I #4 Apr 11 2019 20:59:54
%S 1,0,1,3,3,5,9,14,20,26,38,53,75,101,132,175,229,301,394,509,650,826,
%T 1043,1315,1656,2074,2590,3218,3975,4896,6008,7361,8989,10960,13323,
%U 16159,19531,23553,28323,34002,40723,48694,58115,69249,82350,97766,115832
%N Number of integer partitions of n such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.
%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. The sequence gives the number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 1.
%C The Heinz numbers of these partitions are given by A325185.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.
%e The a(1) = 1 through a(8) = 14 partitions:
%e (1) (21) (22) (41) (51) (61) (71)
%e (31) (311) (321) (322) (332)
%e (211) (2111) (411) (331) (422)
%e (3111) (421) (431)
%e (21111) (511) (521)
%e (3211) (611)
%e (4111) (3221)
%e (31111) (3311)
%e (211111) (4211)
%e (5111)
%e (32111)
%e (41111)
%e (311111)
%e (2111111)
%t otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t Table[Length[Select[IntegerPartitions[n],otb[#]>otb[Rest[#]]&&otb[#]>otb[DeleteCases[#-1,0]]&]],{n,30}]
%Y Cf. A000245, A188674, A325165, A325169, A325183, A325184, A325185, A325187, A325190, A325191.
%K nonn
%O 1,4
%A _Gus Wiseman_, Apr 11 2019
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