cp's OEIS Frontend

A325190 Number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 2.

%I A325190 #6 Feb 16 2025 08:33:58
%S A325190 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 A325190 160,190,214,252,298,364,434,524,620,728,834,966,1112,1306,1522,1788,
%U A325190 2088,2448,2822,3256,3720,4264,4876,5610,6434,7420
%N A325190 Number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 2.
%C A325190 The Heinz numbers of these partitions are given by A325186.
%C A325190 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 A325190   o o o o o o
%C A325190   o o o o o
%C A325190   o o o o o
%C A325190   o o o
%C A325190 with origin-to-boundary graph-distances
%C A325190   4 4 4 3 2 1
%C A325190   3 3 3 2 1
%C A325190   2 2 2 1 1
%C A325190   1 1 1
%C A325190 giving the origin-to-boundary partition (7,5,4,3).
%H A325190 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.
%e A325190 The a(2) = 1 through a(11) = 10 partitions:
%e A325190   (2)   (32)   (33)    (52)     (62)      (72)       (82)        (92)
%e A325190   (11)  (221)  (42)    (22111)  (221111)  (432)      (433)       (443)
%e A325190                (222)                      (3321)     (442)       (533)
%e A325190                (2211)                     (2211111)  (532)       (542)
%e A325190                                                      (3322)      (632)
%e A325190                                                      (3331)      (3332)
%e A325190                                                      (33211)     (33221)
%e A325190                                                      (22111111)  (33311)
%e A325190                                                                  (332111)
%e A325190                                                                  (221111111)
%t A325190 ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];
%t A325190 corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];
%t A325190 Table[Length[Select[IntegerPartitions[n],Apply[Plus,If[#=={},{},FixedPointList[corpos,ptnmat[#]][[-3]]],{0,1}]==2&]],{n,30}]
%Y A325190 Cf. A188674, A325165, A325169, A325183, A325184, A325186, A325187, A325190, A325199.
%K A325190 nonn
%O A325190 0,3
%A A325190 _Gus Wiseman_, Apr 11 2019