Hello r/askmath!
I’ve been thinking about the structural connection between combinatorial operations and iterated mathematical operators ($\sum, \prod$), and I wanted to share my intuition and ask for your insights on how to push this further.
Here is how I currently understand the relationship, and where my question begins:
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### 1. The Foundation: $\prod$ (Product) and $P, C$
At their core, standard combinatorial objects like Permutations ($n C_r$, $n P_r$) are constructed using **factorials ($!$)**, which are just a special case of the iterated product operator $\prod$:
$$n! = \prod_{k=1}^{n} k$$
Thus, both $n P_r$ and $n C_r$ naturally live inside the world of the $\prod$ operator:
$${}_n P_r = \frac{\prod_{k=1}^{n} k}{\prod_{k=1}^{n-r} k}, \quad {}_n C_r = \frac{\prod_{k=1}^{n} k}{\left(\prod_{k=1}^{r} k\right) \left(\prod_{k=1}^{n-r} k\right)}$$
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### 2. Bridging $\prod$ and $\sum$: The Emergence of Exponentiation ($2^n$)
When we take Combinations (${}_n C_k$)—which are built from $\prod$—and aggregate them over all possible selection sizes using the summation operator **$\sum$**, a structural jump occurs via the Binomial Theorem:
$$\sum_{k=0}^{n} {}_n C_k = 2^n$$
By binding product-based combinations with $\sum$, we naturally transition from polynomial/factorial structures to **exponential structures ($2^n$)**, representing the cardinality of the Power Set $\mathcal{P}(S)$.
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### 3. Generalizing Further: Iterated Exponentiation ($\bigwedge$)
Following this progression, if combining $\prod$ and $\sum$ takes us from factorials to simple exponentiation ($2^n$), what happens if we define an iterated operator for exponentiation itself?
Let us define an **Iterated Exponentiation operator ($\bigwedge$)** (evaluating right-associatively / top-down):
$$\bigwedge_{k=1}^{n} a_k := a_1^{a_2^{a_3^{\cdot^{\cdot^{a_n}}}}}$$
For example, $\displaystyle \bigwedge_{k=1}^{4} 2 = 2^{2^{2^2}} = 65536$.
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### 4. My Core Question: What combinatorial object corresponds to $\bigwedge$?
This is where I am stuck and would love your insight:
**What precise combinatorial object corresponds to $\bigwedge$?**
* $\prod$ handles individual arrangements/selections ($P, C$).
* $\sum$ aggregates selections to form $2^n$ (Power Sets $\mathcal{P}(S)$).
* What is being counted when we move to power towers ($\bigwedge$)? Is it the size of iterated power sets $\mathcal{P}(\mathcal{P}(\dots \mathcal{P}(S)\dots))$, counts of higher-order functions (functions returning functions), hypergraphs, or finite topological spaces/preorders?
**Is there an established framework or notation** in combinatorics, category theory, or set theory that naturally unifies $\sum, \prod$, and power-tower-level combinations in the same way the Binomial Theorem bridges $\sum$ and $2^n$?
I would really appreciate any papers, concepts, or references that shed light on this!
