Symmetry reductions and recurrence degrees for banded Toeplitz determinants and permanents

Abstract

This paper studies symmetry-induced reductions of scalar recurrence complexity for balanced banded Toeplitz determinants and permanents. Three mechanisms emerge: determinant state and spectral compression, exceptional permanent--determinant conversion, and symmetries of permanent state spaces. For symmetric determinants, straightening reduces the row-column states from $\binom{2m}{m}$ to the Catalan number $C_{m+1}$; primitive symplectic weight compression then leaves $(3^m+1)/2$ distinct autonomous modes. Skew symmetry has a parallel compound/Hodge explanation: the middle exterior representation splits into two Hodge halves of dimension $\binom{2m}{m}/2$, the one-step compound transfer exchanges the halves, and its two-step restriction has $3^{m-1}$ generic ternary modes. Thus the full skew bound is $2\cdot3^{m-1}$. Widom--Hankel arguments prove generic scalar minimality in both symmetry classes, and every even skew Toeplitz determinant admits an explicit half-size square factorization. For the zero-diagonal pentadiagonal support, a P\'olya--Kasteleyn signing converts every permanent to a determinant. Among consecutive zero-diagonal two-sided bands, universal entrywise conversion---and separately Toeplitz diagonal-wise conversion---occurs only in the Hessenberg families and this pentadiagonal case. Paired renewal identities recover all restored-diagonal determinant and permanent layers. For permanents, transposition gives the open symmetric bound $(\binom{2m}{m}+2^m)/2$, while cyclic defect-sector pairing gives $(4^m+\binom{2m}{m})/2$. Skew-symmetry forces odd-size vanishing and corresponding even-subsequence bounds. In semibandwidth two the open symmetric bound is generically sharp; higher-semibandwidth permanent minimality is left separate from the symmetry reductions proved here.