Some Generalizations of the MacMahon Master Theorem

TL;DR

This paper explores various extensions of the MacMahon Master Theorem by replacing permutations with partial permutations or derangements, broadening its applicability to matrices and submatrices.

Contribution

It introduces new generalizations of the $eta$-extended MacMahon Master Theorem using partial permutations and derangements, expanding its theoretical framework.

Findings

Developed generalized formulas for matrices and submatrices.

Extended the theorem to include partial permutations and derangements.

Provided mathematical proofs for the new generalizations.

Abstract

We consider a number of generalizations of the $\beta$-extended MacMahon Master Theorem for a matrix. The generalizations are based on replacing permutations on multisets formed from matrix indices by partial permutations or derangements over matrix or submatrix indices.