[seqfan] Re: Multiset defining partitions

peter.luschny peter.luschny at googlemail.com
Fri Mar 18 21:48:22 CET 2011


(1)
> is calling "multisets" are apparently equivalence classes of
> multisets, where one multiset can be turned into another by
> "renaming" the elements. It might be nice if there was a simple term
> for such an equivalence class, but I don't know of one.

A good place to look for terminology is the section 7.2.1.5
subsection "partitions of a multiset" in Knuth TAOCP IV.
(fascicle 3, p.74). Also surprising is the fact that the
"Multipartition numbers" P(k,n) as given there on page 141
are unknown (as a table) in OEIS.

(2)
Without using equivalence classes when working with partitions
you effectively bypass the whole systematic of the subject.
I have tried to give a sketch on this topic on my homepage [1].

Many of the earlier sequences of Wolfdieter are based on a special
type of equivalence, equivalence by equal length. Other types of
equivalences correspond to the types

    equal length       -> "len"
    equal biggest part -> "big"
    coarse-grained     -> "sum"
    fine-grained       -> "part"

Only if one looks at these cases side by side on the same generators
one gets an impression of the larger picture. I have compiled some
special cases which go on OEIS under the name generalized Stirling1
and generalized Stirling2 triangles (see [2] and [3]).

(3)
> There should be an initial term 1 with offset 0, representing
> the empty partition.

This is an old problem effecting hundreds of sequences in the
database. The consequences of the basic definitions set by A000070
and A000041 were never enforced.

Related to this is another problem: When I added my share of
triangles to the pool of generalized Stirling triangles I started
with the offset 0 in accordance to A000041. However soon I switched
back to the offset 1 because I wanted the triangles to be displayed
as a triangular array, i.e. I wanted to give them the keyword 'tabl'.

So if consistency (and correctness) as well as the desirable
display as a triangular array is wanted one has to introduce
a new type of triangle (keyword 'tabl0'?)

0           a
1           b
2          c,d
3         e,f,g

[1] http://www.luschny.de/math/seq/CountingWithPartitions.html
[2] http://oeis.org/wiki/User:Peter_Luschny/IndexGeneralStirling1
[3] http://oeis.org/wiki/User:Peter_Luschny/IndexGeneralStirling2

Peter



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