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.

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.

Links

Table of n, a(n) for n=1..82.

Wikipedia, Durfee square.

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.

Cf. A001222, A046660, A051924, A056239, A061395, A093641, A096771, A115994, A243055, A257990, A263297, A325192, A325195.

Sequence in context: A260721 A275318 A190431 * A333452 A190451 A282743

Adjacent sequences: A325175 A325176 A325177 * A325179 A325180 A325181

Keyword

nonn

Author

Gus Wiseman, Apr 08 2019

Status

approved