Real-rooted coefficient permutations

Abstract

Coefficient orders of products of positive linear factors.

Definition 1.1 (Coefficient sequence).

Source. Repository-derived.

Commentary.

For a finite family of real parameters, coeffs r k is the coefficient of X^k in the product of the factors 1 + r i X.

Definition 1.2 (Unimodal permutation).

Source. Repository-derived.

Acknowledgement. T. Ball, D. Galvin, C. Hyry, and K. Weingartner (2022). Independent set and matching permutations. DOI: 10.1002/jgt.22724. URL: https://doi.org/10.1002/jgt.22724.

Commentary.

The inverse positions increase through the value at the final position and decrease after that value.

Definition 1.3 (Weak coefficient ordering).

Source. Repository-derived.

Acknowledgement. T. Ball, D. Galvin, C. Hyry, and K. Weingartner (2022). Independent set and matching permutations. DOI: 10.1002/jgt.22724. URL: https://doi.org/10.1002/jgt.22724.

Commentary.

Arises records a positive parameter family whose nonconstant coefficients are weakly increasing in the order specified by the permutation.

Definition 1.4 (Question 4.6 formal statement).

Source. Repository-derived.

Acknowledgement. T. Ball, D. Galvin, C. Hyry, and K. Weingartner (2022). Independent set and matching permutations. DOI: 10.1002/jgt.22724. URL: https://doi.org/10.1002/jgt.22724.

Commentary.

The claim states that, for every finite size and permutation, weakly sortable coefficient orders are exactly the unimodal permutations.