Let η be a nodal signature, and let σ be a subsignature. Let\[\mathfrak{a} = \left\{ {\begin{array}{*{20}{c}} s & {} & {{\alpha _1}} & {} & {{\alpha _2}} & \cdots & {} & {{\alpha _d}} \\ {} & {{\beta _1}} & {} & {{\beta _2}} & {} & \cdots & {{\beta _d}} & {} \\ \end{array} } \right\}\]be an accordion belonging to the open facet$ \mathcal{S} _{\sigma} $of$\text{C}{{\text{P}}_{\eta }}({{c}_{0}},\cdots ,{{c}_{d}})$. Assuming that$ n|s $we will evaluate$ \Lambda _{ \Gamma } (\mathfrak{a}, \sigma )$.

We will denote\[V(a,b)={{(q-1)}^{a}}{{q}^{(d+1)s-b}},\quad V(a)=V(a,a).\]Let\[ {{\varepsilon _\Gamma }(\sigma ) = {\varepsilon _\Gamma } = \left\{ {\begin{array}{*{20}{c}} {1\quad {\text{if}}\ {\sigma _0} = \square ,} \\ {0\quad {\text{otherwise,}}} \\ \end{array} } \right.} \]\[ \begin{array}{*{20}{c}} {{\mathcal{K}_\Gamma }(\sigma ) = {\mathcal{K}_\Gamma } = \left\{ {i|1 \leqslant i \leqslant d,{\sigma _i} = \square ,{\sigma _{i - 1}} \ne \circ} \right\},\quad {k_\Gamma } = \left| {{\mathcal{K}_\Gamma }} \right|,} \\ {{\mathcal{N}_\Gamma }(\sigma ) = {\mathcal{N}_\Gamma } = \left\{ {i|1 \leqslant i \leqslant d,{\sigma _i} = \square ,{\sigma _{i - 1}} = \circ} \right\},\quad {n_\Gamma } = \left| {{\mathcal{N}_\Gamma }} \right|,} \\ \end{array} \]and\[\begin{array}{*{20}{r}} {{\mathcal{C}_\Gamma }(\sigma ) = {\mathcal{C}_\Gamma } = } \\ {\left\{ {i|1 \leqslant i \leqslant d,{\sigma _0},{\sigma _1}, \cdots ,{\sigma _{i - 1}}\;{\text{not all}}\; \circ\ {\text{and either }}i \in {\mathcal{N}_\Gamma }\;{\text{or}}\;{\sigma _i} = \ast} \right\}}. \\ \end{array} \caption{(15.1)} \]

Let$ c_{\Gamma}=|\mathcal{C}_{\Gamma}| $, and let$ t_{\Gamma} $be the number of$ i $with$ 1 \leqslant i \leqslant d $and$ \sigma _{i} \neq * $. Given a set of indices$\Sigma{=\left\{ {{i}_{1}},\cdots ,{{i}_{k}} \right\}}\subseteq \left\{ 1,2,\cdots ,d \right\}$, let\[{\delta _n}({i_1}, \cdots ,{i_k}) = {\delta _n}(\Sigma ) = {\delta _n}(\Sigma ;\mathfrak{a}) = \left\{ \begin{array}{ll} 1 & {\text{if}}\ n\ {\text{divides}}\ {\alpha _{{i_1}}}, \cdots ,{\alpha _{{i_k}}}, \\ 0 & {\text{otherwise}}. \\ \end{array} \right. \caption{(15.2)}\]Let\[\chi _\Gamma (\mathfrak{a},\sigma ) = \chi _\Gamma = \prod\limits_{i \in\mathcal{C}_\Gamma (\sigma )} {\delta _n (i)} \]. Finally, let\[{a_\Gamma }(\sigma ) = {a_\Gamma } = 2(d - {t_\Gamma } + {n_\Gamma }) + \left\{ \begin{array}{cl} { - 1\quad \mathrm{if}\ {\sigma _0} = \circ } \\ {0\quad \mathrm{if}\ {\sigma _0} = \square } \\ {1\quad \mathrm{if}\ {\sigma _0} = *} \\ \end{array} \right\} + \left\{ {\begin{array}{*{20}{c}} {1\quad \mathrm{if}\ {\sigma _d} = \circ } \\ {0\quad \mathrm{if}\ {\sigma _d} \ne \circ } \\ \end{array} } \right\}.\]

Proposition 15.1*Assume that*$ n|s $.*Given a*Γ-*accordion*\[\mathfrak{a} = \left\{ {\begin{array}{*{20}{c}} s & {} & {{\alpha _1}} & {} & {{\alpha _2}} & \cdots & {} & {{\alpha _d}} \\ {} & {{\beta _1}} & {} & {{\beta _2}} & {} & \cdots & {{\beta _d}} & {} \\ \end{array} } \right\}\]

*and an associated signature*$\sigma \subseteq {{\sigma }_{\mathfrak{T}}}$*not containing the sequence*$ \circ, \square $,*then*

\[{\mathcal{G}_{\Gamma}}(\mathfrak{a},\sigma ) = {( - 1)^{{\varepsilon _\Gamma }}}{\chi _\Gamma } \cdot V({a_\Gamma },{a_\Gamma } + {d_\Gamma }), \caption{(15.3)}\]*where*\[ d_{\Gamma} = \left(\sum_{\begin{smallmatrix}1 \leqslant i \leqslant d\\ \sigma_{i}=\square\end{smallmatrix}} (1 + \delta_{n}(i))\right) + \left\{ \begin{array}{ll} 1 & if \ \sigma_{0} = \square\\ 0 & if \ \sigma_{0} \neq \square \end{array}\right\}. \]

Recall that any subsignature$\sigma$containing the string$\circ\ \square$has...