bibkey: okada2020merged
authors: Okada, So
year: 2020
title: "Merged-log-concavity of rational functions, almost strictly unimodal sequences, and phase transitions of ideal boson-fermion gases"
doi: 10.48550/arXiv.2003.02112
url: "https://arxiv.org/abs/2003.02112v2"
claim: "The paper introduces merged-log-concavity of rational functions, constructs parcels and pre-merged determinants from Gaussian binomial coefficients, relates them to phase transitions of ideal boson-fermion gases, and conjectures that width-three pre-merged determinants with unit weights and base shifts are not almost palindromic."
strata_touched:
  - D5/S3/Combinatorics/GaussianQuotient/OkadaPreMergedNonPalindromic
license: citation-only
triage: anchor

Okada, Merged-log-concavity of rational functions

S. Okada, arXiv:2003.02112v2 (v1 4 March 2020, v2 30 January 2024; primary math.CO, cross-listed cond-mat.stat-mech and hep-th). Quotations are from the v2 TeX source phases.tex.

The pre-parcel of width $\mu$ is $U_a^b(\mu ,w,\rho ,x)={\left[\genfrac{}{}{0ex}{}{b}{a}\right]}_{x^{\rho }}^{w}B(\mu ,w,b-a,\rho ,x)$, with ${\left[\genfrac{}{}{0ex}{}{m}{m^{\prime }}\right]}_x^w={\prod }_i{\left[\genfrac{}{}{0ex}{}{m_i}{m_i^{\prime }}\right]}_{x_i}^{w_i}$, and the pre-merged determinant is $d(U{)}_a^b(\mu ,w,\rho ,e,x)=U_a^b(\mu ,w,\rho ,x)-U_{a^{\vee }}^b(\mu ,w,\rho ,x)\prod x(1,\lambda {)}^e$, where $a^{\vee }$ is the flip of $a$. For $\rho ={\iota }^{\mu }(1)$ the base shift function is $1$. A tuple $(\mu ,a,b)$ is almost strictly pre-fitting if $0\le a_1<\cdots <a_{\mu }\le b_1<\cdots <b_{\mu }$, and $f\in \mathbb{Q}[z]$ is almost palindromic if its nonzero coefficients $f_{{\lambda }_1},\dots ,f_{{\lambda }_{\kappa }}$, listed by increasing degree, satisfy $f_{{\lambda }_i}=f_{{\lambda }_{\kappa -i+1}}$. The paper states:

Let $\mu \in {\mathbb{Z}}_{\ge 3}$. Then, $\text{dt}(U{)}_a^b(\mu ,{\iota }^{\mu }(1),{\iota }^{\mu }(1),0,{\iota }^{\mu }(q))$ is not an almost palindromic $q$-polynomial, provided $\text{tp}\mu ,a,b$ is almost strictly pre-fitting.

With these parameters the polynomial is ${\prod }_i{\left[\genfrac{}{}{0ex}{}{b_i}{a_i}\right]}_q-{\prod }_i{\left[\genfrac{}{}{0ex}{}{b_i}{a_{\mu +1-i}}\right]}_q$.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2003.02112
• Source: https://arxiv.org/abs/2003.02112v2