cp's OEIS Frontend

A325184 Last part of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

%I A325184 #5 Feb 16 2025 08:33:58
%S A325184 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 A325184 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 A325184 3,1,19,1,2,1,20,2,21,1,1,1,4,1,22,1,3,1
%N A325184 Last part of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.
%C A325184 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 A325184 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A325184 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.
%e A325184 The partition with Heinz number 7865 is (6,5,5,3), with diagram
%e A325184   o o o o o o
%e A325184   o o o o o
%e A325184   o o o o o
%e A325184   o o o
%e A325184 with origin-to-boundary graph-distances
%e A325184   4 4 4 3 2 1
%e A325184   3 3 3 2 1
%e A325184   2 2 2 1 1
%e A325184   1 1 1
%e A325184 giving the origin-to-boundary partition (7,5,4,3) with last part 3, so a(7865) = 3.
%t A325184 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325184 ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];
%t A325184 corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];
%t A325184 Table[Apply[Plus,If[n==1,{},FixedPointList[corpos,ptnmat[primeptn[n]]][[-3]]],{0,1}],{n,100}]
%Y A325184 Positions of 1's are A325185. Positions of 2's are A325186.
%Y A325184 Cf. A056239, A065770, A093641, A112798, A174090, A325166, A325169, A325183, A325187.
%K A325184 nonn
%O A325184 1,3
%A A325184 _Gus Wiseman_, Apr 08 2019