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Maximum of different products of partitions of n into distinct parts.
4

%I #46 Oct 28 2023 11:43:21

%S 1,1,2,3,4,6,8,12,15,24,30,40,60,72,120,144,180,240,360,420,720,840,

%T 1008,1260,1680,2520,2880,5040,5760,6720,8064,10080,13440,20160,22680,

%U 40320,45360,51840,60480,72576,90720,120960,181440,201600,362880,403200

%N Maximum of different products of partitions of n into distinct parts.

%H Alois P. Heinz, <a href="/A034893/b034893.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..1000 from T. D. Noe)

%H Tomislav Doslic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Doslic/doslic15.html">Maximum product over partitions into distinct parts</a>, J. of Integer Sequences, Vol. 8 (2005), Article 05.5.8.

%H Andrew V. Sills and Robert Schneider, <a href="https://arxiv.org/abs/1904.08004">The product of parts or "norm" of a partition</a>, arXiv:1904.08004 [math.NT], 2019. Also in <a href="http://math.colgate.edu/~integers/u8/u8.Abstract.html">Integers</a>, (2020) Vol. 20A, Article #A13.

%e 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

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,

%p `if`(n=0, 1, max(b(n, i-1), i*b(n-i, min(n-i, i-1)))))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 19 2019

%t Table[Max[Times@@@Select[IntegerPartitions[n],Max[Tally[#][[All,2]]]<2&]],{n,50}] (* _Harvey P. Dale_, May 28 2017 *)

%t 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]]]]];

%t a[n_] := b[n, n];

%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Aug 21 2019, after _Alois P. Heinz_ *)

%Y Cf. A000792, A309859 (corresponding partitions).

%K nonn,nice

%O 0,3

%A _Erich Friedman_

%E a(0)=1 prepended by _Alois P. Heinz_, Apr 19 2019