Let $C$ be collection of subsets of $[N]$. Given, $n\in [N]$, what's the terminology for $card\{s\subseteq [N]\mid n\not \in s,\, s \cup\{n\}\in C\}$?

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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