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%I #4 Apr 10 2019 22:01:55
%S 0,1,2,2,3,1,4,3,1,1,5,1,6,1,2,4,7,2,8,1,2,1,9,1,2,1,2,1,10,1,11,5,2,
%T 1,3,2,12,1,2,1,13,1,14,1,1,1,15,1,3,1,2,1,16,3,3,1,2,1,17,1,18,1,1,6,
%U 3,1,19,1,2,1,20,2,21,1,1,1,4,1,22,1,3,1
%N Last part of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.
%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.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.
%e The partition with Heinz number 7865 is (6,5,5,3), with diagram
%e o o o o o o
%e o o o o o
%e o o o o o
%e o o o
%e with origin-to-boundary graph-distances
%e 4 4 4 3 2 1
%e 3 3 3 2 1
%e 2 2 2 1 1
%e 1 1 1
%e giving the origin-to-boundary partition (7,5,4,3) with last part 3, so a(7865) = 3.
%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%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[Apply[Plus,If[n==1,{},FixedPointList[corpos,ptnmat[primeptn[n]]][[-3]]],{0,1}],{n,100}]
%Y Positions of 1's are A325185. Positions of 2's are A325186.
%Y Cf. A056239, A065770, A093641, A112798, A174090, A325166, A325169, A325183, A325187.
%K nonn
%O 1,3
%A _Gus Wiseman_, Apr 08 2019
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