On the Number of Many-to-Many Alignments of Multiple Sequences

TL;DR

This paper develops formulas to count the number of alignments among multiple sequences based on specified match-up types, providing new asymptotic results for particular cases.

Contribution

It introduces a novel counting method for sequence alignments with specified match-up constraints and derives an asymptotic formula for a specific set of match types.

Findings

Derived a new asymptotic formula for alignments with match types in {1,2}.

Counted the number of nonnegative integer matrices with fixed row sums and column constraints.

Provided a comprehensive combinatorial framework for sequence alignment enumeration.

Abstract

We count the number of alignments of $N \ge 1$ sequences when match-up types are from a specified set $S\subseteq \mathbb{N}^N$. Equivalently, we count the number of nonnegative integer matrices whose rows sum to a given fixed vector and each of whose columns lie in $S$. We provide a new asymptotic formula for the case $S=\{(s_1,\ldots,s_N) \:|\: 1\le s_i\le 2\}$.