bibkey: alofi2024lackingbipartite authors: Amal Alofi, Mark Dukes year: 2024 title: “A note on the lacking polynomial of the complete bipartite graph” doi: 10.1016/j.disc.2024.114323 url: https://arxiv.org/abs/2411.02667v1 claim: “For the stochastic sandpile model of Chan, Marckert and Selig, the paper characterises the stochastically recurrent states of K_{2,n} and K_{m,n} with n = 2, bounds n^(m-1) m^(n-1) <= |Sto(K_{m,n})| <= n^(m-1) m^n, and asks in Question 6 whether |Sto(K_{m,n})| is larger or smaller than n^(m-1) m^n / 2.” strata_touched:

• D5/S3/StatisticalMechanics/Sandpiles/StochasticSandpileBipartiteHalf license: citation-only triage: anchor

Alofi and Dukes, the lacking polynomial of the complete bipartite graph

In the stochastic sandpile model a stable configuration assigns to each non-sink vertex v a number of grains 0 ≤ c(v) < d(v). An orientation O of G is compatible with c when

in_O(v) ≥ d(v) − c(v)

at every non-sink vertex (Definition 1, following Chan, Marckert and Selig), and Sto(G) is the union over all orientations of the compatible stable configurations (Theorem 2). On K_{m,n} the sink v_0 lies in the part {v_0, …, v_{m−1}}. From the spanning-tree count and the number of stable configurations the paper records

n^{m−1} m^{n−1} ≤ |Sto(K_{m,n})| ≤ n^{m−1} m^n

and asks:

Question 6. Can it be determined whether or not the number of stochastically recurrent states dominates the set of stable states? I.e. can it be decided |Sto(K_{m,n})| ≶ n^{m−1} m^n / 2 ?

Verified locator

• DOI: 10.1016/j.disc.2024.114323 (Crossref record retrieved 2026-09-27: Discrete Mathematics 348(2) (2025), article 114323; authors Alofi, Dukes).
• URL: https://arxiv.org/abs/2411.02667v1 (the only version listed by the arXiv API on 2026-09-27); source file ssm_final.tex: stable configurations (lines 158–162), Definition 1 defcom (lines 164–173), Theorem 2 thcom (lines 177–181), the graph K_{m,n} (lines 194–197), Example 5 (lines 222–328), the bounds (lines 330–340) and Question 6 (lines 342–345).