/partitions-negative-parts-vertex-decompositions

A Haskell module exporting five functions that implement enumeration mechanisms described in Theorem 6.4 from the monograph A.O. Matveev, Symmetric Cycles, Jenny Stanford Publishing, 2023, and illustrated in Example 6.5.

Primary LanguageHaskellMIT LicenseMIT

Partitions of the Negative Parts of Vertices of Hypercube Graphs, and Vertex Decompositions w.r.t. Distinguished Symmetric Cycles

A Haskell module exporting five functions that implement enumeration mechanisms described in Theorem 6.4 from the monograph A.O. Matveev, Symmetric Cycles, Jenny Stanford Publishing, 2023, and illustrated in Example 6.5.

Let $t$ be an integer, $t\geq 3$. We denote by $[t]$ the ground set $\langle 1,2,\ldots, t\rangle$. Given odd integers $\ell',\ell'',\ell\in [t]$, and positive integers $j'$ and $j''$, we are interested in statistics related to the following family of ordered pairs $(A,B)$ of disjoint unordered subsets $A$ and $B$ of the ground set $[t]$:

$\langle(A,B) \in \boldsymbol{2}^{[t]} \times \boldsymbol{2}^{[t]}:\ \ |A\cap B|=0,\ \ \ 0\neq|A|=j',\ \ \ 0\neq|B|=j'',\ \ \ A\dot\cup B\subsetneqq [t],\ \ \ \mathfrak{q}(A)=\ell',\ \ \ \mathfrak{q}(B)=\ell'',\ \ \ \mathfrak{q}(A\dot\cup B)=\ell\rangle.$

Here the quantity $\mathfrak{q}(A):=|\boldsymbol{Q}({}_{-A}\mathrm{T}^{(+)},\boldsymbol{R})|$ means the size of the decomposition set of the vertex, whose negative part is $A$, w.r.t. the distinguished symmetric cycle $\boldsymbol{R}$ in the hypercube graph on its vertex set $\langle 1,-1\rangle^t$; see a note in the distinguished-symmetric-cycles repository.