# some properties of the partitions of n

Wouter Meeussen eu000949 at pophost.eunet.be
Sun Oct 18 13:56:12 CEST 1998

```hi,

if we look at the partitions of a number as all the ways such number can be
written as a sum,
eq:

Partitions[7]

{{7},
{6,1},
{5,2},
{5,1,1},
{4,3},
{4,2,1},
{4,1,1,1},
{3,3,1},
{3,2,2},
{3,2,1,1},
{3,1,1,1,1},
{2,2,2,1},
{2,2,1,1,1},
{2,1,1,1,1,1},
{1,1,1,1,1,1,1}}

we could ask how these pieces could be 'reassembled' in a different way.
A simple way to reassemble would be to multiply them :

Apply[Times,Partitions[7], 1 ]
Out[]=
{7,6,10,5,12,8,4,9,12,6,3,8,4,2,1}

at this stage, two properties come foreward: how many different results are
there, and, what is the largest one?

ser3=Table[Length[Union[Apply[Times, Partitions[n], 1 ]]],{n,30}]
Out[]=
{1,2,3,4,6,8,11,14,18,23,29,36,45,55,67,81,98,117,140,166,196,231,271,317,369,
429,496,573,660,758}
(is not in EIS)

Table[Max[Union[Apply[Times, Partitions[n], 1 ]]],{n,30}]
Out[]=
{1,2,3,4,6,9,12,18,27,36,54,81,108,162,243,324,486,729,972,1458,2187,2916,
4374,6561,8748,13122,19683,26244,39366,59049}

this is known to EIS in a completely different (??) context :
----------------------
%I A000792 M0568 N0205
%S A000792 1,2,3,4,6,9,12,18,27,36,54,81,108,162,243,324,486,729,972,1458,
%T A000792 2187,2916,4374,6561,8748,13122,19683,26244,39366,59049,78732,118098
%N A000792 a(3n)=3^n, a(3n+1)=4.3^{n-1}, a(3n+2)=2.3^n.
%R A000792 CMB 8 627 1965. JRM 4 168 1971. FQ 27 16 1989.
%O A000792 1,2
%A A000792 njas
%K A000792 nonn
----------------------
further, we could try to weed out the multiple occurrences of low factors by
replacing the multiplication with LeastCommonMultiple LCM:

again, the maximum of the LCM's and a count of'm:

Table[Max[Union[Apply[LCM, Partitions[n], 1 ]]],{n,30}]
Out[]=
{1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,210,210,420,420,420,420,840,840,
840,1260,1260,1540,2310,2520,4620}

is known in EIS as:
----------------------
%I A000793 M0537 N0190
%S A000793
1,1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,210,210,420,420,420,420,
%T A000793 840,840,1260,1260,1540,2310,2520,4620,4620,5460,5460,9240,9240,13860,
%U A000793 13860,16380,16380,27720,30030,32760,60060,60060,60060,60060,120120
%N A000793 Landau's function g(n): largest order of permutation of n elements.
%R A000793 BSMF 97 187 1969.
%O A000793 1,3
%D A000793 J.-L. Nicolas, pp. 228ff of R. L. Graham et al., eds.,
Mathematics of Paul Erdos I.
%A A000793 njas
%K A000793 nonn
%E A000793 Extended by David Wilson.
---------------------

Table[Length[Union[Apply[LCM, Partitions[n], 1 ]]],{n,30}];
Out[]=
{1,2,3,4,6,6,9,11,14,16,20,23,27,31,35,43,47,55,61,70,78,88,98,111,123,136,
152,168,187,204}

is known in EIS as :
----------------------
%I A009490
%S A009490
1,1,2,3,4,6,6,9,11,14,16,20,23,27,31,35,43,47,55,61,70,78,88,98,111,123,
%T A009490
136,152,168,187,204,225,248,271,296,325,356,387,418,455,495,537,581,629,
%U A009490 678,732,787,851,918,986,1056,1133,1217,1307,1399,1498,1600,1708,1823
%N A009490 Number of distinct orders of permutations of n objects.
%O A009490 0,3
%K A009490 nonn
%A A009490 wilson at ctron.com
----------------------

I do not know in how far these 'amendments' are usefull,
or how to format such ammendments, so I'll format them
pretentiously as new results (they aren't of course) :

in EIS Format:

%I A000001
%S A000001 1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,210,210,420,420,420,
420,
%T A000001 840,840,840,1260,1260,1540,2310,2520,4620
%N A000001 largest LCM of the partitions of n
%R A000001
%Y A000001 cf. A000002
%A A000001 Wouter Meeussen, w.meeussen.vdmcc at vandemoortele.be
%O A000001 0,2
%t A000001 Table[Max[Union[Apply[LCM, Partitions[n], 1 ]]],{n,30}]
%K A000001 nonn

%I A000002
%S A000002 1,2,3,4,6,6,9,11,14,16,20,23,27,31,35,43,47,55,61,70,78,88,98,111,
%T A000002 123,136,152,168,187,204
%N A000002 number of different LCM's of the partitions of n
%R A000002
%Y A000002 cf. A000001
%A A000002 Wouter Meeussen, w.meeussen.vdmcc at vandemoortele.be
%O A000002 0,2
%t A000002 Table[Length[Union[Apply[LCM, Partitions[n], 1 ]]],{n,30}]
%K A000002 nonn

%I A000003
%S A000003 1,2,3,4,6,8,11,14,18,23,29,36,45,55,67,81,98,117,140,166,196,231,
271,
%T A000003 317,369,429,496,573,660,758
%N A000003 number of different products of the partitions of n
%R A000003
%Y A000003 cf. A000004
%A A000003 Wouter Meeussen, w.meeussen.vdmcc at vandemoortele.be
%O A000003 0,2
%t A000003 Table[Length[Union[Apply[Times, Partitions[n], 1 ]]],{n,30}]
%K A000003 nonn

%I A000004
%S A000004 1,2,3,4,6,9,12,18,27,36,54,81,108,162,243,324,486,729,972,1458,
2187,
%T A000004 2916,4374,6561,8748,13122,19683,26244,39366,59049
%N A000004 maximum of the different products of the partitions of n
%R A000004
%Y A000004 cf. A000003
%A A000004 Wouter Meeussen, w.meeussen.vdmcc at vandemoortele.be
%O A000004 0,2
%t A000004 Table[Max[Union[Apply[Times, Partitions[n], 1 ]]],{n,30}]
%K A000004 nonn
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be

```