On the permanent of Sylvester-Hadamard matrices

TL;DR

This paper proves a conjecture relating the permanent of Sylvester-Hadamard matrices to the dyadic valuation of factorials, showing these permanents are non-zero for all orders n ≥ 2.

Contribution

It establishes a precise relationship between the permanent of Sylvester-Hadamard matrices and dyadic valuations, confirming the non-vanishing property for all relevant sizes.

Findings

Dyadic valuation of the permanent equals that of n! for all n ≥ 2

Permanent of Sylvester-Hadamard matrices does not vanish for n ≥ 2

Confirms Wanless's conjecture for these matrices

Abstract

We prove a conjecture due to Wanless about the permanent of Hadamard matrices in the particular case of Sylvester-Hadamard matrices. Namely we show that for all n greater or equal to 2, the dyadic valuation of the permanent of the Sylvester-Hadamard matrix of order n is equal to the dyadic valuation of n!. As a consequence, the permanent of the Sylvester-Hadamard matrix of order n doesn't vanish for n greater or equal to 2.