---
title: Permutational 1-11-Representation Number
url: https://www.emergentmind.com/topics/permutational-1-11-representation-number
type: topic
---

# Permutational 1-11-Representation Number

A permutational 1-11-representation number is a graph invariant defined via word representations of graphs where adjacency is encoded by constraints on consecutive factors in concatenations of permutations. Given a finite simple undirected graph $G=(V,E)$, the permutational 1-11-representation number $\pi_{11}(G)$ (or $R_{\pi}(G)$) is the minimum number $k$ such that there exists a word $w = P_1 P_2 \cdots P_k$, with each $P_i$ a permutation of $V$, and $w$ is a 1-11-representation of $G$—meaning that for each pair of distinct vertices $x,y$, adjacency is determined by the number of repeated consecutive letters in the restricted word $w|_{\{x,y\}}$. This concept is central to research on combinatorial word representations of graphs and is closely linked to regular languages and absolute Stirling numbers of the first kind [2601.20490][1805.11445].

## 1. Formal Definitions and Core Properties

Let $G=(V,E)$ be a finite simple undirected graph. Define $V^+$ as the set of all nonempty words over the alphabet $V$. For $w \in V^+$ and $X \subseteq V$, the restriction $w|_X$ is defined as the word formed by deleting from $w$ all letters not in $X$.

A word $w\in V^+$ is a **1-11-representation** of $G$ if for every distinct $x,y\in V$,
$$(x,y)\in E \iff \text{in } w|_{\{x,y\}} \text{, at most one occurrence of } xx\text{ and at most one occurrence of } yy.$$
Equivalently, $x$ and $y$ are non-adjacent precisely when $w|_{\{x,y\}}$ contains either two factors $xx$, or two $yy$, or one of each.

A **permutational 1-11-representation** of $G$ is a 1-11-representation $w$ that is a concatenation of permutations of $V$: $w = P_1 P_2 \cdots P_k$. The **permutational 1-11-representation number** $\pi_{11}(G)$ is the minimum $k$ for which such a representation exists [2601.20490].

## 2. Cube-Free and Square-Free Representation Phenomena

A **cube** in a word is a factor of the form $XXX$ for some nonempty $X$. A central result for permutational 1-11-representations establishes that any cube in $w$ can always be eliminated without changing the encoded graph. Specifically, if $w = P_1P_2\cdots P_p$ is a permutational 1-11-representation of $G$ containing a cube $XXX$, one can delete one entire copy of $X$ (or, if $X$ is a single permutation, delete two consecutive identical permutations) to obtain a shorter valid representation. The principal corollary is that any shortest permutational 1-11-representation (one achieving $\pi_{11}(G)$) is guaranteed to be cube-free [2601.20490].

By contrast, squares (factors $XX$) may be unavoidable even in optimal-length permutational 1-11-representations. For example, for $G = K_3 \sqcup \{v\}$, $\pi_{11}(G) = 3$, but every such representation with three permutations necessarily contains a square, and no square-free construction is possible at this length [2601.20490].

## 3. Tabulation and Computation for Small Values

For graphs $G$ where $|V|=n\leq 11$, the permutational 1-11-representation numbers are explicitly connected to the signless Stirling numbers of the first kind, $s(n,k)$. These numbers can be extracted from the coefficients in the expansion of the rising factorial $P_n(m) = m(m+1)\cdots (m+n-1)$ in the monomial basis, via the lower-triangular permutation-generation matrix $C_n$ and its inverse $B_n$ [1805.11445].

Table: Absolute Stirling Numbers of the First Kind $s(n,k)$ for $n=1$ to $n=5$

| $n$ | $s(n,1)$ | $s(n,2)$ | $s(n,3)$ | $s(n,4)$ | $s(n,5)$ |
|---|---|---|---|---|---|
| 1 | 1 |   |   |   |   |
| 2 | 1 | 1 |   |   |   |
| 3 | 2 | 3 | 1 |   |   |
| 4 | 6 | 11 | 6 | 1 |   |
| 5 | 24 | 50 | 35 | 10 | 1 |

The complete triangle up to $n=11$ follows the recurrence $s(n,k)=s(n-1,k-1)+(n-1)\,s(n-1,k)$, with boundary values $s(n,1)=(n-1)!$, $s(n,n)=1$. The row sums yield $n!$ for each $n$ [1805.11445].

## 4. Bounds, Examples, and Complexity

Every graph $G$ admits a permutational 1-11-representation, so $1\leq \pi_{11}(G)<\infty$. For cliques, $\pi_{11}(K_n)=1$ since any permutation suffices. If $G$ has at least one non-edge, then $\pi_{11}(G)\geq 2$. For the disjoint union $K_3 \sqcup \{v\}$, $\pi_{11}(G)=3$. However, no efficient general formula is known for computing $\pi_{11}(G)$, and determining its exact value is a hard combinatorial optimization problem with open complexity status [2601.20490].

A trivial upper bound arises from the fact that every graph on $n$ vertices is 2-11-representable by some concatenation of permutations, giving $\pi_{11}(G) \leq R(G) \leq 2n(n-1)/2$, but this is not tight in practice [2601.20490].

## 5. Regularity and Automata-Theoretic Structure

For a fixed graph $G=(V,E)$, the set of all 1-11-representations forms a regular language over $V$. For each unordered pair $\{a,b\} \subseteq V$, the relevant sublanguage
$$L_{a,b} = \{w \in V^* \mid \text{in } w|_{\{a,b\}}, a \text{ and } b \text{ each appear at least once, with at most one } aa \text{ and at most one } bb\}$$
is regular and recognized by a small DFA. The overall language of 1-11-representations is:
$$L(G) = \bigcap_{\{a,b\} \in E} L_{a,b} \cap \bigcap_{\{a,b\} \notin E} \overline{L_{a,b}},$$
hence is regular [2601.20490]. The set of permutational 1-11-representations $Perm(G) = L(G) \cap (Perm(V))^+$ is also regular, since the set of all permutations of $V$ is finite and regular. This regularity enables direct construction of DFAs for recognition and algorithmic search for minimum-length representations, with recognition complexity $O(|w||V|^2)$ [2601.20490].

## 6. Connections to Combinatorics and Algebra

The triangle of permutational 1-11-representation numbers coincides with the absolute Stirling numbers of the first kind. These numbers enumerate permutations on $n$ elements with a specified number of cycles. They appear in combinatorics in diverse settings, including the expansions of falling and rising factorial polynomials and the conversion between product-polynomial and monomial bases. In the context of permutational 1-11-representations, they facilitate the analysis of representation numbers for small $n$ and structure the associated coefficient matrices [1805.11445].

The relevant matrices $C_n$ and $B_n = C_n^{-1}$, with explicit recurrences, provide algebraic tools for manipulating the combinatorial invariants and extracting the required enumeration for $n \leq 11$. The key generating function is
$$G\{P_n(m)\}(x) = \sum_{m=1}^\infty P_n(m)x^m = \frac{n! x}{(1-x)^{n+1}}.$$
This links the power sums with the structure of permutational representations and their enumeration [1805.11445].

Source: https://www.emergentmind.com/topics/permutational-1-11-representation-number