bibkey: lillo2026prioritylattice
authors: Adrián Lillo and Mercedes Rosas
year: 2026
title: The Priority Lattice
doi: 10.48550/arXiv.2603.28905
url: https://arxiv.org/abs/2603.28905v1
claim: "Section 6, item 3 asks for isomorphic principal intervals and conjectures left-factorial and quadratic counts; item 4 asks whether every priority lattice is sublattice dismantlable."
strata_touched:
  - D5/S3/Combinatorics/PriorityLattice/PrincipalFilters
  - D5/S3/Combinatorics/PriorityLattice/PrincipalIdeals
  - D5/S3/Combinatorics/PriorityLattice/SublatticeDismantling
license: citation-only
triage: anchor

Principal intervals of the priority lattice

Verified locator

DOI: 10.48550/arXiv.2603.28905

Source: https://arxiv.org/abs/2603.28905v1

Definitions

Section 2.1, pp. 3–4:

a priority forest (labeled with $[n{]}_0$ and with $m$ edges) is a rooted forest $(T_0,T_1,\dots ,T_{n-m})$ with all component trees increasing and ordered according to the root’s labels, and where for all $j<k$, every label in $T_j$ is smaller than every label in $T_k$. Thus, the vertex set of each tree is an interval of integers

Section 3.1, p. 4:

The underlying set of $\Pi (n)$ consists of all priority forests labeled with $[n{]}_0$, together with an extra element $\hat{1}$. The order relation $\le$ on $\Pi (n)$ is defined as follows. Given two priority forests $P$ and $P^{\prime }$, $P\le P^{\prime }$ if and only if $E(P)\subseteq E(P^{\prime })$. On the other hand, $\hat{1}$ is set to be the top element of $\Pi (n)$.

A parent function on Fin (n + 1) records a root by none and a non-root by its unique smaller parent. Relation.EqvGen records connected components, and their convexity records the interval condition. The forest order is edge inclusion; WithTop adjoins the extra greatest element.

Final-section question

Section 6, item 3, pp. 24–25:

How many principal ideals of $\Pi (n)$ are isomorphic to $\Pi (m)$, for some $1\le m\le n$? Let ${\gamma }_n$ denote the number of principal ideals of $\Pi (n)$ that are isomorphic to $\Pi (m)$ with $m\le n$. The sequence $({\gamma }_n{)}_{n\ge 1}$ begins as $2,4,8,14,22,32,44,\dots$ This sequence seems to be https://oeis.org/A014206, which is given by the formula ${\gamma }_n=n^2+n+2.$ We could also modify this question and ask the number ${\theta }_n$ of principal filters of $\Pi (n)$ (this is, intervals of the form $[P,\hat{1}]$) that are isomorphic to $\Pi (m)$ with $m\le n$. This sequence begins as $2,4,10,34,154,874,\dots$ and seems to be https://oeis.org/A003422, the sequence of left factorials, given by ${\theta }_n={\sum }_{i=0}^{n-1}k!.$

Counting conventions and conclusions

The filter data give ${\theta }_n={\sum }_{k=0}^{n}k!$, the left factorial $!(n+1)$ (OEIS A003422). The printed sum has mismatched indices and its upper bound disagrees with the displayed data.

The ideal data give ${\gamma }_n=n^2-n+2$, OEIS A014206 with its index shifted by one. The displayed $n^2+n+2$ is false for every $n\ge 1$. Allowing only $1\le m\le n$ deletes the $n$ ideals isomorphic to $\Pi (0)$ and gives $n^2+2-2n$. In natural-number arithmetic this is encoded as n ^ 2 + 2 - 2 * n.

The counts include the principal ideal of the greatest element. Section 3, p. 7 says, “We do not consider $\Pi (n)$ to be a principal ideal”. Excluding that ideal decreases each ideal count by one. The delivered counts follow the source’s numerical data and the convention in issue 14857, including its natural-subtraction correction.

A principal filter qualifies precisely when every component except the last is a singleton. Its last tree contracts to give the required order isomorphism. An ideal below a forest has rank equal to the edge count and at most that many lower covers. Comparing with the factorial coatom count forces the target parameter to be at most two. Compression to closed edge subsets then enumerates the possible shapes.

Computed evidence and open boundary

The independent experiment entry is priority-lattice intervals. Its check.py 6 command returned exit code 0; the program SHA-256 is b71ca8894f15984e5eca4712f947556f024d3d2ddfc7ebe68142090fc7f562ec. This is finite computed evidence, distinct from the general Lean proofs.

The source’s questions on the flag quasi-symmetric function and a maximal-chain map to minimal cycle factorizations remain open. The disjoint-partition reading of sublattice dismantlability is refuted at n = 6 by SublatticeDismantling.result; its dossier gives the exact scope. The enumeration does not alter the source’s other proved results, which do not require these conjectural interval counts. Classifying intervals with both endpoints proper remains open.

Absence of every prior published settlement remains ASSUMED-UNVERIFIED; the bounded literature search in issue 14857 found none. The general interval enumerations are repository-derived conclusions, while the source definitions and increasing-tree/permutation correspondence are literature background.

Sublattice dismantling question

Section 6, item 4:

A sublattice dismantling is a subdivision of a lattice into two non-empty sublattices. A lattice is sublattice dismantlable if it can be decomposed into 1-element lattices by consecutive sublattice dismantlings. We have checked with the help of SageMath that $\Pi (n)$ is sublattice dismantlable for $n=$ 1 to 5. Is this always the case?

The standard subdivision reading is a disjoint partition, as in the identical Sage definition and the interval/complement implementation at immutable commit 133eb21d995e6aa5b6de2cd8db798a442e5cc033. This identifies the convention; it does not identify the authors’ historical Sage version or establish their intended semantics beyond the source wording. An overlapping-cover interpretation would be a different question.

Under this convention, sixteen explicit source forests and the adjoined top form a seventeen-element ambient sublattice of $\Pi (6)$ isomorphic to the incidence lattice of $K_{3,3}$. It has no partition into two nonempty sublattices. Restriction of a recursive dismantling to a nonempty sublattice, discarding empty branches, gives a contradiction. SublatticeDismantling.result therefore refutes the all-positive-n assertion on the existing literal Pi carrier. Ambient meets and joins for these elements are proved from edge inclusion against arbitrary bounds, without relying on an unrestricted general formula. The count is sixteen forests plus top; “seventeen forests plus top” is not the construction. No worldwide priority claim is made.