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Let $M$ be a positive homogeneously presented cancellative monoid ${< L \midR >}_{mo}$ equipped with the degree map $\deg:M \to \Z_{\ge0}$ defined byassigning to each equivalence class of words the length of the words, and let$P_{M, \deg}(t):= \sum_{u \in\ M}t^{\deg(u)}$ be its generating series, calledthe growth function. If $M$ satisfies the condition that any subset $J$ of$I_0$ ($:=$ the image of the set $L$ in $M$) admits either the least rightcommon multiple $\Delta_{J}$ or no common multiple in $M$, then the inversionfunction $P_{M, \deg}(t)^{-1}$ is given by the polynomial $\sum_{J \subsetI_{0}}(-1)^{#J} t^{\deg(\Delta_{J})}$, where the summation index $J$ runs overall subsets of $I_0$ whose least right common multiple exists. Since a monoid$M$ generally may not admit the least right common multiple $\Delta_{J}$ for agiven subset $J$ of it, if we attempt to generalize the formula, theconsideration to obtain the above formula is invalid. To resolve thisobstruction, we will examine the set $\mathrm{mcm}(J)$ of minimal common rightmultiples of $J$. Then, we need to introduce a concept of a tower of minimalcommon multiples of elements of $M$ and denote the set of all the towers in $M$by $\mathrm{Tmcm}(M)$. Considering the structure of the set $\mathrm{Tmcm}(M)$,K. Saito has proved the inversion formula \[ P_{M,\deg}(t). N_{M,\deg}(t)=1, \]where the second factor in LHS is a suitably signed generating series \[N_{M,\deg}(t):= 1 + \sum_{T\in\mathrm{Tmcm}(M)}(-1)^{#J_1+...+#J_{n}-n+1}\sum_{\Delta\in \mathrm{mcm}(J_n)}t^{\deg(\Delta)}, \] called the skew growth function. In this article, wepresent several explicit calculations of examples of the skew growth functionsfor the monoid of type $\mathrm{B_{ii}}$ and others whose towers do not stop onthe first stage $J_1$.

The skew growth functions $N_{M, \mathrm{deg}}(t)$ for the monoid of type $\mathrm{B_{ii}}$ and others