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Let $N \ge 1$ be an integer and let $\mathfrak S$ be a nonempty collection of subsets of $[N] := \{1,2,\ldots,N\}$. For any $n \in [N]$, define $\partial_n \mathfrak S := \{S\setminus\{n\} \mid S \in \mathfrak S\} = \{A \subseteq [N] \mid n \not \in A\text{ and }A \cup \{n\} \in \mathfrak S\}$. Also, define $d_n := |\partial_n \mathfrak S|$, $\overline d := \max_{n \in [N]} d_n$, $\underline d := \min_{n \in [N]} d_n$.

Question. Is there any geometric / topoligical meaning or terminology for $\partial_n \mathfrak S$, $d_n$, $\overline d$, or $\underline d$ ?

Examples

  • If $d \in [1,N]$ is an integer and $\mathfrak S = K_{N,d}$ is the collection of subsets of $[N]$ with exactly $d$ elements, then $\partial_n \mathfrak S$ is isomorphic to $K_{N-1,d-1}$, and so $d_n = \overline d = \underline d = |K_{N-1,d-1}| = {N-1 \choose d-1}$ for all $n \in [N]$.

  • If $D \in [1,N]$ is an integer and $\mathfrak S = K_{N,\le D} = \cup_{d=0}^D K_{N,d}$ is the collection of subsets $[N]$ with $D$ elements or fewer, then $\partial_n \mathfrak S$ is isomorphic to $K_{N-1,D-1}$ and so $d_n = \overline d = \underline d = |K_{N-1,\le D-1}| = \sum_{d=0}^{D-1} {N-1 \choose d}$ for all $n \in [N]$.

  • If $G_1,\ldots,G_k$ are a partitioning of $[N]$ and $\mathfrak S$ is the collection of subsets $A$ of $[N]$ such that $A$ contains exactly one element of each $G_i$, then $\partial_n \mathfrak S$ is isomorphic to $\prod_{i \ne i(n)} G_i$ (where $i(n)$ is the index of $G_i$ which contains $n$) and so $d_n = \prod_{i \ne i(n)} |G_i|$, $\overline d = (\prod_i N_i) / \min_i N_i$, and $\underline d := (\prod_i N_i) / \max_i N_i$. Note that $\mathfrak S$ also corresponds to the collection of all complete sets of representatives for the equivalence relationship $n \sim n'$ iff $n$ and $n'$ belong to the same $G_i$, i.e iff $i(n) = i(n')$.

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