An upper bound for the type of a numerical semigroup, and a reduction of Wilf's conjecture

Let $S$ be a numerical semigroup with multiplicity $\text{mult}$, conductor $\text{cc}$, embedding dimension $\text{ee}$, type $\text{typ}$ and genus $\text{gnus}$, and let $\text{nn}=\text{cc}-\text{gnus}$. Wilf’s conjecture asserts that $\text{ee}\text{nn}\ge \text{cc}$; the inequality $\text{gnus}\le \text{typ}\text{nn}$ of Fr"oberg, Gottlieb and H"aggkvist settles it when $\text{typ}\le \text{ee}-1$. The Ap'ery set of $S$ with respect to any $s\in S\setminus \{0\}$ carries a partial order whose maximal elements are the pseudo-Frobenius numbers translated by $s$; for $s=\text{mult}$ its minimal elements are the minimal generators other than $\text{mult}$. Comparing the two extremal statistics bounds the type by $\text{typ}\le \text{ee}-1+\text{Xii}(S)\le \text{ee}-1+\mathrm{\Theta }(S)$, where $\mathrm{\Theta }(S)$ measures the redundancy of the covering of the gaps of $S$ by the pseudo-Frobenius numbers and $\text{Xii}(S)$ refines it. With an exact decomposition of the Wilf number this yields the genus bound $\text{gnus}\le \text{ee}-1+\text{typ}(\text{nn}-1)$, strictly stronger than $\text{gnus}\le \text{typ}\text{nn}$ precisely when $\text{typ}\ge \text{ee}$, and reduces Wilf’s conjecture to an inequality free of $\text{cc}$ and $\text{nn}$. We determine the equality case of $\text{gnus}\le \text{typ}\text{nn}$, recovering a classification of Singhal; answer a question of Moscariello and Sammartano whenever $\text{ee}\ge \text{typ}+1$; and correct Kaplan’s classification of the equality case for $\text{cc}\le 2\text{mult}$, from which an infinite family is missing.