# Computational Aspects of Ordered Integer Partitions with Bounds

## Abstract

This paper is dedicated to the counting problem of writing an integer number z as a sum of an ordered sequence of n integers from n given intervals, i.e., counting the number of configurations $$(z_1,\ldots ,z_n)$$ with $$z = z_1 + \cdots + z_n$$ for $$z_i \in [x_i, y_i]$$ with integers $$x_i$$ and $$y_i$$ and $$1 \le i \le n$$. We show an algorithm computing this number in $$\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n z^{\lg n}\right)$$ average time, and a data structure computing this number in $$\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n\right)$$ time, independently of z. The data structure is constructed in $$\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( 2^n n^3\right)$$ time. Its construction algorithm only depends on the intervals $$[x_i,y_i]$$ ($$1 \le i \le n$$). This construction algorithm can be parallelized with $$\pi = \mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n^3\right)$$ processors, yielding $$\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( 2^n \frac{n^3}{\pi }\right)$$ construction time with high probability.

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## Notes

1. 1.

In particular, we allow negative integer values.

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## Acknowledgements

This work is partly funded by the JSPS KAKENHI Grant Number JP18F18120.

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Correspondence to Dominik Köppl.

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Parts of this work have already been presented at the 12th International Symposium on Experimental Algorithms [10].

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