The theme on this page is to upgrade classical theorems from analysis to the realm of symmetric functions. Several of these classical theorems assert pointwise positivity. It is then natural to ask if they can then be upgraded to coefficientwise positivity.
Let \({\bf X} = (x_i)_{i\geq 0} \), \({\bf Y} = (y_i)_{i\geq 0} \) and \({\bf Z} = (z_i)_{i\geq 0} \) be three sequences of indeterminates.
(P3.1.a) Conjecture: All the Toeplitz minors of the sequence \((e_n({\bf X} ) e_n({\bf Y} ))_{n\geq 0}\) are monomial-positive in \({\bf X} \) and \({\bf Y} \).
(P3.1.b) Conjecture: All the Toeplitz minors of the sequence \((n! \; e_n({\bf X} ) e_n({\bf Y} ))_{n\geq 0}\) are monomial-positive in \({\bf X} \) and \({\bf Y} \).
(P3.1.c) Conjecture: All the Toeplitz minors of the sequence \((n!^{p} \; e_n({\bf X} ) e_n({\bf Y} ) e_n({\bf Z} ))_{n\geq 0}\) are monomial-positive in \({\bf X} \), \({\bf Y} \) and \({\bf Z} \) for \(p=0,1,2\).
For \(k\geq 2\) and \(1\leq m\leq k\) consider sequence of indeterminates \({\bf X^{(m)}} = (x_i^{(m)})_{i\geq 0} \).
Conjecture: For \(p = 0,1,\ldots,k-1\), all the Toeplitz minors of the sequence \((n!^p \; \prod_{m=1}^k e_n({\bf X^{(m)}} ))_{n\geq 0}\) are monomial-positive in all indeterminates.
For Conjecture P3.1, and more generally for Conjecture P3.2 at \(p=0\), Angarone, Kim, Oh and Soskin (2025) showed that certain minors indexed by skew shapes not containing any \(3\times 2\) block of cells are not only monomial positive, but also Schur positive.
Conjecture: Fix \(M\geq 1\). Then the Toeplitz minors of the sequence \(\left(\prod_{i=1}^M (a_i + b_i n) e_n({\bf X})\right)_{n\geq 0}\) are monomial-positive in \({\bf X}\), coefficientwise in the \(a_i\) and \(b_i\).
An AI-generated solution to Conjecture P3.3 for the affine case of \(M=1\) has been posed by an anonymous author here: https://evidencepress.org/releases/affine-elementary-symmetric-total-nonnegativity/. This solution is yet to be verified.
Conjecture: The Toeplitz minors of the sequence \(\left(c_n e_n({\bf X})\right)_{n\geq 0}\) are monomial-positive in \({\bf X}\), whenever \((c_n)_{n\geq 0}\) are real numbers with \(\sum_{n=0}^\infty \dfrac{c_n}{n!} t^n \in LP^{+}\).