This Maple worksheet accompanies the papers: 

Di Nardo E., G. Guarino, D. Senato (2008), A Maple algorithm for polykays and their generalizations, Adv. Appl. Stat. Vol. 8, No. 1, 19 - 36, http://www.pphmj.com/journals/adas.htm.

Di Nardo E., G. Guarino, D. Senato (2008), An unifying framework for k-statistics, polykays and their generalizations, Bernoulli. Vol. 14(2), 440-468. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, http://isi.cbs.nl/bernoulli/, (download from http://www.unibas.it/utenti/dinardo/lavori.html)
Di Nardo E., G. Guarino, D. Senato (2008), Symbolic computation of moments of sampling distributions, Comp. Stat. Data Analysis Vol. 52, no. 11, 4909-4922, (download from http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.0129v1.pdf or http://www.unibas.it/utenti/dinardo/lavori.html)
Multiset subdivisionE. Di Nardo*
elvira.dinardo@unibas.it
http://www.unibas.it/utenti/dinardo/home.html;
Tel: +39 0971205890, Fax: +39 0971205896G. Guarino**
giuseppe.guarino@asl2.potenza.itD. Senato*
domenico.senato@unibas.it  
* Dipartimento di Matematica e Informatica, Universit? degli Studi della Basilicata,Viale dell'Ateneo Lucano n.10, 85100 Potenza, Italy**Medical Scool, Universit? del Sacro Cuore (Rome branch), Largo Agostino Gemelli n.8, 00168 Roma, Italy 

Introduction 

Abstract: .The algorithm allows us to build subdivisions of multiset, reducing the overall computational complexity using the integer partitions. 

Application Areas/Subject: Combinatorics & algebraic and combinatorial methods 

Keyword: Set partitions, multiset 

See Also:  For applications see [3]  

Initialization 

 

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(2.1)

 

Background to Subdivisions of a Multiset 

Consruction of set partitions 

List all partitions of a set. The parameter "N" indicates the number of blocks in which the set is subdivided.  

 

 

Example 

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(3.1.1)

 

 

Subdivision of a special multiset:  

When the multiset has support equal to a {}, the output is the follow: 

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	(3.2.1)

 

 

 

The output can be compacted by using the following notation where the subdivisions appear togeter with their multiplicity: 

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The above print-out is the output of the following routine: 

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Example 

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(3.2.2)

 

Subdivisions of a multiset 

When in the multiset there are two or more different elements, the result is different from the previous one. Also in this case we use integer partitions for constructing subdivisions. This device reduces the computational complexity.
The list of all partitions of a set with 3 blocks is:  

,, , , . 

Setting    we obtain: 

 

Compacting the previous output we obtain: 

 

 

The Maple routines 

Some details on secondary Maple routines 

The following function calcules the product of factorials of multiplicity. 

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Example 

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(4.1.1.1)

 

 

The following function inserts an element into an array with the following rules:  

1) the element must not be in a previous block with the same or less degree 

2) jump every block equal to the previous  

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Examples 

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(4.1.1.2)

 

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(4.1.1.3)

 

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(4.1.1.4)

 

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(4.1.1.5)

 

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(4.1.1.6)

 

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The following function inserts more then one elements into an array with the same rules.  

Note: the array contains also the multeplicity of  the multiset 

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Examples 

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(4.1.1.7)

 

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(4.1.1.8)

 

 

 

The following function is an extension of previous one. The input can be more then two vectors. 

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Examples 

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(4.1.1.9)

 

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(4.1.1.10)

 

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The following function makes the Cartesian Product. 

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Examples 

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(4.1.1.11)

 

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(4.1.1.12)

 

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The master function 

 

A Maple algorithm for listing multisets subdivisions. 

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Examples 

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(4.1.2.1)

 

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(4.1.2.2)

 

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(4.1.2.3)

 

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Note: the notion of subdivision as been introduced in [2].  

In the subdivision list, the sum of  all multiplicity is the Bell Number.  

Moreover the same multiplicities of the subdivision with the same block numbers is the Stirling Number of the second kind. 

Example: from  "makeTab(4)" we obtain:   

 

in wich it is possible to observe:  

1 = combinat['stirling2'](4,4) =combinat['stirling2'](4,1); 

6 = combinat['stirling2'](4,3); 

3 + 4 = 7 = combinat['stirling2'](4,2); 

and 1 + 6 + 3 + 4 +  1 = 15 = combinat['Bell'](4). 

 

Conclusions 

The multiset subdivision is very usefull in speed-up the algorithm that generates power sum from augumented symmetric functions and vice cersa. So it can be used also in the more general theory of simmetric functions. We ave used the multiset subdivision for calcolate k-statistics, polykays and their generalizations (see references). 

References 

[1] Di Nardo E., G. Guarino, D. Senato (2008) A Maple algorithm for polykays and their generalizations. Adv. Appl. Stat. Vol. 8, No. 1, 19 - 36, http://www.pphmj.com/journals/adas.htm.

[2] Di Nardo E., G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. Vol. 14(2), 440-468. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, http://isi.cbs.nl/bernoulli/, (download from http://www.unibas.it/utenti/dinardo/lavori.html)
 

[3]  Di Nardo E., G. Guarino, D. Senato, A Maple algorithm for k-statistics, polykays and their multivariate generalization, source Maple algorithm located in www.maplesoft.com (submitted) 

[4] Di Nardo E., G. Guarino, D. Senato (2008) Symbolic computation of moments of sampling distributions. Comp. Stat. Data Analysis Vol. 52, no. 11, 4909-4922, (download from http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.0129v1.pdf or http://www.unibas.it/utenti/dinardo/lavori.html) 

 

 

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