A257990 The side-length of the Durfee square of the partition having Heinz number n.

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1

Offset: 1

Author

Emeric Deutsch, May 18 2015

Keywords

• nonn

Comments

The Durfee square of a partition is the largest square that fits inside the Ferrers board of the partition.

We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.

In the Maple program the subprogram B yields the partition with Heinz number n.

First appearance of k is a(prime(k)^k) = k. - Gus Wiseman, Apr 12 2019

Examples

			a(9)=2; indeed, 9 = 3*3 is the Heinz number of the partition [2,2] and, clearly its Durfee square has side-length =2.
		

References

• G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass. 1976.
• G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
• M. Bona, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Findstat, St000183: The side length of the Durfee square of an integer partition

Crossrefs

Positions of 1's are A093641. Positions of 2's are A325164. Positions of 3's are A307386.

Cf. A006918, A056239, A062457, A065770, A112798, A115720, A117485, A215366, A252464, A325163, A325169.

Programs

Maple

with(numtheory): a := proc (p) local B, S, i: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: S := {}: for i to nops(B(p)) do if i <= B(p)[nops(B(p))+1-i] then S := `union`(S, {i}) else  end if end do: max(S) end proc: seq(a(n), n = 2 .. 146);
# second Maple program:
a:= proc(n) local l, t;
      l:= sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`);
      for t from nops(l) to 1 by -1 do if l[t]>=t then break fi od; t
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, May 10 2016

Mathematica

a[n_] := a[n] = Module[{l, t}, l = Reverse[Sort[Flatten[Table[PrimePi[ f[[1]] ], {f, FactorInteger[n]}, {f[[2]]}]]]]; For[t = Length[l], t >= 1, t--, If[l[[t]] >= t, Break[]]]; t]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Feb 17 2017, after Alois P. Heinz *)

Formula

For a partition (p_1 >= p_2 >= ... > = p_r) the side-length of its Durfee square is the largest i such that p_i >=i.

Extensions

a(1)=0 prepended by Alois P. Heinz, May 10 2016