some properties of the partitions of n

Wouter Meeussen eu000949 at
Sun Oct 18 13:56:12 CEST 1998


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

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 ]

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}]
 (is not in EIS)

Table[Max[Union[Apply[Times, Partitions[n], 1 ]]],{n,30}]

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}]

is known in EIS as:
%I A000793 M0537 N0190
%S A000793
%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}];

is known in EIS as :
%I A009490
%S A009490
%T A009490
%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

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,
%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
%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
%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,
%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
%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,
%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
%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
eu000949 at

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