Annali di Matematica Pura ed Applicata

, Volume 106, Issue 1, pp 329–351

# Enumeration of weighted rectangular arrays

• Margaret J. Hodel
Article

## Summary

Let Ip(n, k; q1, q2, ..., qp)=Ip(n, k) be defined by
$$I_v (n,k) = \sum {\prod\limits_{i = 1}^v {q_i^{j\mathop = \limits^{\mathop \sum \limits^n a_{ij} } 1} } }$$
, where the summation is over all p-line arrays of positive integers {aij:1≤i≤p, 1≤j≤n} subject to the following conditions:
$$\begin{array}{*{20}c} {\max \left\{ {a_{ij} :1 \leqslant i \leqslant p} \right\} \leqslant \min \left\{ {a_{i,j + 1} :1 \leqslant i \leqslant p} \right\}, 1 \leqslant j \leqslant n - 1,} \\ {\max \left\{ {a_{ij} :1 \leqslant i \leqslant p} \right\} \leqslant j, 1 \leqslant j \leqslant n,} \\ {a_{i + 1,j} \leqslant a_{ij} , 1 \leqslant i \leqslant p - 1, 1 \leqslant j \leqslant n,} \\ \end{array}$$
and
$$a_{1j} = a_{2j} = ... = a_{pj} for k values of j, 1 \leqslant j \leqslant n$$
. Assuming$$\prod\limits_{i = 1}^p {q_i = 1}$$, formulas for Iv(n, k) and another closely related enumerant are determined in this paper. These two functions generalize enumerants which Carlitz has obtained.

### Keywords

Positive Integer Rectangular Array Related Enumerant

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### References

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