Counting in logarithmic space

2026-07-17

AI Summary

This work investigates which combinatorial and number-theoretic counting functions are computable in logarithmic space, i.e., belong to the complexity class #L. By developing a framework that counts accepting paths of nondeterministic logspace Turing machines and integrating tools from combinatorial encoding, discrete geometry, and representation theory, the study systematically establishes the #L-computability of numerous classical functions. Key contributions include the first unified inclusion of Catalan numbers, Stirling numbers, and the number of standard Young tableaux within #L; proofs that multinomial coefficients, linear extensions of trees, and GL₂-plethysm coefficients under bounded outer partitions lie in #L or are verifiable in log² space; and a novel conditional approach to refuting their #P-completeness, thereby substantially expanding the theoretical frontier of low-complexity counting problems.

Abstract

We study the class $\#\mathsf{L}$ of functions counting accepting paths of non-deterministic log-space Turing machines and construct methods to prove containment in $\#\mathsf{L}$. We prove that a large number of classical combinatorial and number theoretic functions belong to this class: classical functions from enumerative combinatorics (multinomial coefficients, Catalan numbers, linear extensions of trees, Stirling numbers, etc), algebraic combinatorics (number of standard Young tableaux, etc), discrete geometry, number theoretic functions, representation theoretic multiplicities in a large class of cases. We show that $\mathrm{GL}_2$-plethysm coefficients of bounded length outer partition can be counted by log$^2$-space polytime verifiers. We pose numerous questions and conjectures on $\#\mathsf{L}$ containment and its generalizations, that suggest venues for conditionally disproving $\#\mathsf{P}$-completeness. While studying which combinatorial functions are in $\#\mathsf{P}$ provides a formal way of (dis)proving the existence of combinatorial interpretations, the lower class $\#\mathsf{L}$ serves as an analogue for functions computable in polynomial time.

Problem

• #L
• logarithmic space
• counting functions
• combinatorial enumeration
• computational complexity

Innovation

• #L
• logarithmic space
• combinatorial functions
• plethysm coefficients
• counting complexity