## A Class of Multinomial Permutations Avoiding Object Clusters

**Authors:** Richard J. Mathar

The multinomial coefficients count the number of ways (of permutations) of
placing a number of partially distinguishable objects on a line, taking ordering
into account. A well-known two-parametric family of counts arises if there are objects
of c distinguishable colors and m objects of each color, m*c objects in total, to be placed on line.
In this work we propose an algorithm to count the permutations where
no two objects of the same color appear side-by-side on the line. This eliminates
all permutations with "clusters" of colors. Essentially we represent filling
the line sequentially with objects as a tree of states where each node
matches one partially filled line. Subtrees are merged if they have the same
branching structure, and weights are assigned to nodes in the tree keeping track
of how many mergers take place. This is implemented in a JAVA program; numerical results
confirm Hardin's earlier counts for this kind of restricted permutations.

**Comments:** Pages 9 to 21 are a JAVA program distributed under the LGPL v3.

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### Submission history

[v1] 2015-11-02 15:32:57

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