Real stability of layer-refined antichain polynomials for three-chain products with a two-element factor

Abstract

For all positive integers $n,k$, we prove that the layer-refined antichain polynomial of the product poset $[2]\times[n]\times[k]$ is real stable. Jacobi-polynomial interlacing further shows that its diagonal specialization, the ordinary antichain polynomial of the same poset, has only simple, strictly negative zeros. For the special family $[2]\times[m]\times[m+1]$, explicit reciprocal identities give palindromicity; reciprocal pairing of the simple negative zeros then shows that every coefficient in the gamma expansion is strictly positive. Thus we prove Conjecture 4.3 of Ding and Dong and resolve all parts of their Conjecture 4.5, while strengthening its stated gamma-positivity consequence. The enumerative input is an explicit first-crossing reflection for two lattice paths, specialized from work of Krattenthaler and Sulanke.