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A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

%I A325178 #8 Apr 10 2019 22:01:35
%S A325178 0,0,1,1,2,1,3,2,0,2,4,2,5,3,1,3,6,1,7,2,2,4,8,3,1,5,1,3,9,1,10,4,3,6,
%T A325178 2,2,11,7,4,3,12,2,13,4,1,8,14,4,2,1,5,5,15,2,3,3,6,9,16,2,17,10,2,5,
%U A325178 4,3,18,6,7,2,19,3,20,11,1,7,3,4,21,4,2,12
%N A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.
%C A325178 The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.
%C A325178 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%D A325178 Richard P. Stanley, Enumerative Combinatorics, Volume 2,  Cambridge University Press, 1999, p. 289.
%H A325178 Wikipedia, <a href="https://en.wikipedia.org/wiki/Durfee_square">Durfee square</a>.
%F A325178 a(n) = A263297(n) - A257990(n).
%e A325178 The partition (3,3,2,1) has Heinz number 150 and diagram
%e A325178   o o o
%e A325178   o o o
%e A325178   o o
%e A325178   o
%e A325178 containing maximal square
%e A325178   o o
%e A325178   o o
%e A325178 and contained in minimal square
%e A325178   o o o o
%e A325178   o o o o
%e A325178   o o o o
%e A325178   o o o o
%e A325178 so a(150) = 4 - 2 = 2.
%t A325178 durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
%t A325178 codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
%t A325178 Table[codurf[n]-durf[n],{n,100}]
%Y A325178 Positions of zeros are A062457. Positions of 1's are A325179. Positions of 2's are A325180.
%Y A325178 Cf. A001222, A046660, A051924, A056239, A061395, A093641, A096771, A115994, A243055, A257990, A263297, A325192, A325195.
%K A325178 nonn
%O A325178 1,5
%A A325178 _Gus Wiseman_, Apr 08 2019