[seqfan] Re: An Arrangement Of Partitions

[Leroy Quet]

Thanks to Franklin for his quick response.

It was just pointed out to me that it is customary to list the parts of a partition in non-increasing order, not in non-decreasing order. So, I have just submitted the proper arrangement as table A175025.

(I guess someone should submit a table that is as A175023, but reading the lengths of the runs of each term of A175020 from RIGHT TO LEFT instead. I'll let someone else do that... I am too tired now to do it myself.)

Thanks,
Leroy Quet

[ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] ) ]
>....
> %I A175023
> %S A175023 
> 1,1,1,2,1,2,1,1,1,3,1,3,1,2,1,1,1,1,1,2,2,4,1,4,1,3,1,1,2,1,1,1,2,2,1,
> %T A175023 1,1,1,1,2,3,5
> %N A175023 Irregular table read by rows: Row n (of
> A175022(n) terms) 
> contains
> the run-lengths in the binary representation of A175020(n),
> reading 
> left to
> right.
> %C A175023 This table lists the parts of the partitions of
> the positive
> integers. Each partition is represented exactly once in
> this table. If 
> n is such
> that 2^(m-1) <= A175020(n) <= 2^m -1, then row n of
> this table gives 
> one
> partition of m.
> %e A175023 Table to start:
> %e A175023 1
> %e A175023 1,1
> %e A175023 2
> %e A175023 1,2
> %e A175023 1,1,1
> %e A175023 3
> %e A175023 1,3
> %e A175023 1,2,1
> %e A175023 1,1,1,1
> %e A175023 2,2
> %e A175023 4
> %e A175023 1,4
> %e A175023 1,3,1
> %e A175023 1,2,1,1
> %e A175023 1,2,2
> %e A175023 1,1,1,1,1
> %e A175023 2,3
> %e A175023 5
> %e A175023 Note there are: 1 row that sums to 1, two rows
> that sum to 
> 2, three
> rows that sum to 3, five rows that sum to 4, seven rows
> that sum to 5, 
> etc,
> where 1,2,3,5,7,... are the number of unrestricted
> partitions of 
> 1,2,3,4,5,...
> %Y A175023 Cf. A175020,A175022,A175024
> %K A175023 base,more,nonn,tabf
> %O A175023 1,4
> 
> (A175021 contains those positive integers not in A175020.
> And A175024 
> is the
> table A175023 with the terms of each row arranged in
> non-descending 
> order.)
>....