A034893 Maximum of different products of partitions of n into distinct parts.

1, 1, 2, 3, 4, 6, 8, 12, 15, 24, 30, 40, 60, 72, 120, 144, 180, 240, 360, 420, 720, 840, 1008, 1260, 1680, 2520, 2880, 5040, 5760, 6720, 8064, 10080, 13440, 20160, 22680, 40320, 45360, 51840, 60480, 72576, 90720, 120960, 181440, 201600, 362880, 403200

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from T. D. Noe)

Tomislav Doslic, Maximum product over partitions into distinct parts, J. of Integer Sequences, Vol. 8 (2005), Article 05.5.8.

Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019. Also in Integers, (2020) Vol. 20A, Article #A13.

Example

The partitions of n = 4 are (4), (1, 3), (2, 2), (1, 1, 2) and (1, 1, 1, 1) with the products of partitions being 4, 3, 4, 2 and 1 respectively. As these are 4 distinct numbers (being 1, 2, 3 and 4) we have a(4) = 4. - David A. Corneth, Apr 28 2020

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
     `if`(n=0, 1, max(b(n, i-1), i*b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 19 2019

Mathematica

Table[Max[Times@@@Select[IntegerPartitions[n], Max[Tally[#][[All, 2]]]<2&]], {n, 50}] (* Harvey P. Dale, May 28 2017 *)
b[n_, i_] := b[n, i] = If[i(i+1)/2<n, 0, If[n==0, 1, Max[b[n, i-1], i b[n-i, Min[n-i, i-1]]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 21 2019, after Alois P. Heinz *)

Crossrefs

Cf. A000792, A309859 (corresponding partitions).

Sequence in context: A081029 A300787 A171966 * A187448 A321266 A018662

Adjacent sequences: A034890 A034891 A034892 * A034894 A034895 A034896

Keyword

nonn,nice

Author

Erich Friedman

Extensions

a(0)=1 prepended by Alois P. Heinz, Apr 19 2019

Status

approved