bibkey: elder2024toggling authors: Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, Amanda Welch year: 2024 title: “Toggling, rowmotion, and homomesy on interval-closed sets” doi: 10.48550/arXiv.2307.08520 url: https://arxiv.org/abs/2307.08520v2 claim: “Conjecture 4.9 states that the number of maximal elements minus the number of minimal elements is 0-mesic under rowmotion on interval-closed sets of every product of two finite chains.” strata_touched:

• D5/S3/Combinatorics/Geometry/RectangleRowmotionHomomesy
• D5/S3/Combinatorics/Geometry/IntervalClosedSignedCardinalityRefutation license: citation-only triage: anchor

Toggling, rowmotion, and homomesy on interval-closed sets

The paper introduces interval-closed sets and their toggle rowmotion. Conjecture 4.9 states that, for every product of two finite chains, the number of maximal members minus the number of minimal members is 0-mesic on every rowmotion orbit. The repository target retains the literal interval-closed objects and resolves this exact conjecture.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2307.08520
• URL: https://arxiv.org/abs/2307.08520v2
• Version and location: arXiv:2307.08520v2, Conjecture 4.9 (the conjecture is also identified in the paper’s product-of-chains discussion).

Signed cardinality

Conjecture 4.12 (p. 29) reads: “If m = 2 or m = 3, then the signed cardinality statistic is 0-mesic under rowmotion on interval-closed sets of [m]×[n] whenever m + n − 1 is even.”

Definition 3.17 (p. 21) reads: “Fix a finite poset P. For each x ∈ P, define the signed cardinality statistic SC(x): P → {−1, 1} as follows:” with SC(x) = 1 if rk(x) is even and −1 if rk(x) is odd. It continues: “For an interval-closed set I, SC(I) = ∑_{x∈I} SC(x).”

Definition 2.6 (p. 4) defines t_x(I) = I − {x} if x ∈ I and I − {x} ∈ IC(P), and t_x(I) = I otherwise; if x ∉ I it defines t_x(I) = I ∪ {x} if I ∪ {x} ∈ IC(P), and t_x(I) = I otherwise. Its final sentence is: “That is, x is toggled in/out of I if doing so results in another interval-closed set.” Definition 2.9 (p. 5) reads: “Given an interval-closed set I ∈ IC(P), the rowmotion of I, Row(I), is given by applying all toggles in the reverse order of any linear extension.”

The signed-cardinality refutation uses the literal toggles and the distinct forward orbit on [3]×[12]. The seed {(1,7),(3,2),(3,3),(3,4),(3,5)} has a 73-state orbit with signed-cardinality sum −1. The separate max-minus-min statistic and its all-rectangle result are different statements.