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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.
14
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, 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, 4, 3, 18, 6, 7, 2, 19, 3, 20, 11, 1, 7, 3, 4, 21, 4, 2, 12
OFFSET
1,5
COMMENTS
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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
REFERENCES
Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.
FORMULA
a(n) = A263297(n) - A257990(n).
EXAMPLE
The partition (3,3,2,1) has Heinz number 150 and diagram
o o o
o o o
o o
o
containing maximal square
o o
o o
and contained in minimal square
o o o o
o o o o
o o o o
o o o o
so a(150) = 4 - 2 = 2.
MATHEMATICA
durf[n_]:=Length[Select[Range[PrimeOmega[n]], Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1, 0, Max[PrimeOmega[n], PrimePi[FactorInteger[n][[-1, 1]]]]];
Table[codurf[n]-durf[n], {n, 100}]
CROSSREFS
Positions of zeros are A062457. Positions of 1's are A325179. Positions of 2's are A325180.
Sequence in context: A260721 A275318 A190431 * A333452 A190451 A373891
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 08 2019
STATUS
approved