The computation of Kostka Numbers and Littlewood-Richardson Coefficients is #P-complete

TL;DR

This paper proves that computing Kostka numbers and Littlewood-Richardson coefficients is #P-complete, establishing their computational difficulty and connecting it to known #P-complete problems in combinatorics.

Contribution

It demonstrates the #P-completeness of calculating Kostka numbers and Littlewood-Richardson coefficients, resolving a longstanding folklore question.

Findings

Computing Kostka numbers is #P-complete.

Computing Littlewood-Richardson coefficients is #P-complete.

Reductions are from contingency table counting problems.

Abstract

Kostka numbers and Littlewood-Richardson coefficients play an essential role in the representation theory of the symmetric groups and the special linear groups. There has been a significant amount of interest in their computation. The issue of their computational complexity has been a question of folklore, but was asked explicitly by E. Rassart. We prove that the computation of either quantity is #P-complete. The reduction to computing Kostka numbers, is from the #P-complete problem of counting the number of 2 x k contingency tables having given row and column sums. The main ingredient in this reduction is a correspondence discovered by D. E. Knuth. The reduction to the problem of computing Littlewood-Richardson coefficients is from that of computing Kostka numbers.