Ultra-Log-Concavity from a Matrix Recursion

Definitions and mathematical statement

Let $z$ be a formal indeterminate. Define $T_n,Q_n,C_n\in{\mathbb{Z}}_{\geq0}[z]$, for integers $n\geq4$, by \[ \begin{pmatrix}T_4\\Q_4\\C_4\end{pmatrix} =\begin{pmatrix}z^4+2z^3\\z^4\\2z^4+2z^3\end{pmatrix},\qquad \begin{pmatrix}T_{n+1}\\Q_{n+1}\\C_{n+1}\end{pmatrix} =\begin{pmatrix} z&1+z&1+z\\ 0&1+z&z\\ z+z^2&0&1+z \end{pmatrix} \begin{pmatrix}T_n\\Q_n\\C_n\end{pmatrix}. \] Set \[ V_n(z)=T_n(z)+Q_n(z)+C_n(z),\qquad d_n=\deg V_n,\qquad v_{n,k}=[z^k]V_n(z)\quad(0\leq k\leq d_n). \] For a finite nonnegative sequence $(a_k)_{k=0}^{d}$, its support has no internal zeros if $a_i,a_j>0$ and $i<j$ imply $a_k>0$ for every $i<k<j$. The sequence is log-concave if \[ a_k^2\geq a_{k-1}a_{k+1}\qquad(1\leq k\leq d-1). \] It is ultra-log-concave of order $d$ if its support has no internal zeros and the sequence $\left(a_k/\binom{d}{k}\right)_{k=0}^{d}$ is log-concave; explicitly, \[ \left(\frac{a_k}{\binom{d}{k}}\right)^2 \geq \frac{a_{k-1}}{\binom{d}{k-1}} \frac{a_{k+1}}{\binom{d}{k+1}} \qquad(1\leq k\leq d-1). \] A finite sequence is unimodal if there is an index $m$ such that $a_0\leq\cdots\leq a_m\geq\cdots\geq a_d$.

Conjecture attributed to Germain Poullot. For every integer $n\geq4$, the coefficient sequence $(v_{n,k})_{k=0}^{d_n}$ is log-concave, with the stronger ultra-log-concavity assertion requested by the cleaned registry statement. Under the explicit convention of ultra-log-concavity of order $d_n$, that stronger assertion is that the support has no internal zeros and \[ \left(\frac{v_{n,k}}{\binom{d_n}{k}}\right)^2 \geq \frac{v_{n,k-1}}{\binom{d_n}{k-1}} \frac{v_{n,k+1}}{\binom{d_n}{k+1}} \quad\text{for every }1\leq k\leq d_n-1. \] Unresolved normalization. The supplied excerpt uses "(ultra-)log-concave" without defining the ultra-log-concavity order or specifying whether initial zero coefficients are removed and the sequence reindexed. The order-$d_n$ formulation above is a precise candidate interpretation requiring confirmation, not a uniquely specified source convention. The excerpt writes the sequence index as $n$, whereas the cleaned statement takes log-concavity in $k$ for each fixed $n$; the latter interpretation is used here explicitly.

The source places this recursion in the study of coherent monotone paths of the hypersimplex \[ \Delta(n,2)=\left\{x\in[0,1]^n:\sum_{i=1}^n x_i=2\right\}. \] It reports a counting interpretation for the coefficients and computational checks of the conjecture through $n=300$. The supplied excerpt does not specify the path statistic indexed by $k$, the monotonicity data, or which precise ultra-log-concavity convention was checked; these claims are retained as source context, not additional verified hypotheses or results.

Broader research request. Develop criteria ensuring unimodality and ultra-log-concavity for coefficient sequences arising from polynomial-vector recursions with nonnegative polynomial matrix entries. The record does not specify a class of initial vectors, matrix dimensions, an ultra-log-concavity order, or whether the desired conclusion concerns individual components, their sum, or other combinations. This part therefore remains a request for a theory rather than a fully quantified conjecture.

Short English statement

For each integer $n\geq4$, form the three recursively defined polynomials and add them. The conjecture asks for log-concavity, and more strongly ultra-log-concavity, of the resulting coefficients as $k$ varies. The normalization for ultra-log-concavity needs confirmation. A broader request seeks criteria for similar polynomial-matrix recursions.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30005718$, title "Ultra-Log-Concavity from a Matrix Recursion," exact source problem code catalogue entry. The supplied original excerpt identifies "Problem 6 (Germain Poullot)," although the structured proposer field is empty.
• Supplied report citation: Geometric, Algebraic and Topological Combinatorics (2024), Oberwolfach Reports, pp. 3249--3318, DOI 10.4171/owr/2023/58. Supplied source URL and literature-source link labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2023/58. The registry separately gives proposed year 2023. These year labels are preserved without resolving their relationship.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim; no independent source inspection is asserted in this draft.
• Literature assessment dated 2026-08-22: the supplied status is "open." This is supplied dated curation, not independently verified. Its assessment discusses realizability of M-convex functions and Pluecker vectors of subspace arrangements, stating that the primary report leaves both questions open. Those are different questions from the matrix-recursion conjecture, so this assessment does not establish the latter's literature status. The supplied background itself calls for independent verification of time-sensitive status and cited claims.
• The supplied original excerpt reports computational checks through $n=300$, gives a coherent-monotone-path interpretation, and ends mid-sentence. Its initial questions about realizable M-convex functions and subspace-arrangement Pluecker vectors appear before the heading for Poullot's problem; their relationship to this record is unresolved.