Cardinalities of Kiselman monoids and their parity asymptotics

Abstract

The cardinalities of Kiselman's semigroups $K_n$ are known to grow double exponentially in $n$: the sequence $2^{-n/2}\log_2|K_n|$ converges along the even and odd $n$ subsequences to positive limits $c_{\mathrm{even}}$ and $c_{\mathrm{odd}}$ that appear to coincide. In this paper we prove that they are indeed distinct. Since Kiselman's semigroups are $\mathcal{L}$-trivial, the suffix relation defines a partial order on $K_n$. We introduce the \emph{Kiselman transform} $\mathsf{Kis}(P)$ of a poset $P$, a construction recovering the poset associated with $K_{n+1}$ from that associated with $K_n$. Counting strictly increasing chains in $P$ and $\mathsf{Kis}(P)$ yields a linear recurrence governed by a symmetric infinite matrix with explicit binomial entries. This can be used towards efficiently computing $|K_n|$, allowing the values listed in OEIS sequence A125625 to be extended beyond $n=14$. A generating functions analysis gives formulas for the above asymptotic limits, together with rigorous numerical bounds establishing that \[ c_{\mathrm{even}} - c_{\mathrm{odd}} = 1.9945066605\ldots\times10^{-6} \]

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