Permutational 1-11-Representation Number

• Permutational 1-11-Representation Number is a graph invariant defined by concatenating vertex permutations to represent adjacency with 1-11 constraints.
• Optimal representations are cube-free while squares may be inevitable, highlighting complexity in minimizing permutation concatenations.
• The invariant connects with absolute Stirling numbers of the first kind and regular language theory, facilitating combinatorial and automata-theoretic analysis.

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 (Das et al., 28 Jan 2026, Zhu, 2018).

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 (Das et al., 28 Jan 2026).

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 (Das et al., 28 Jan 2026).

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 (Das et al., 28 Jan 2026).

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 $B_n$ (Zhu, 2018).

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 $n$ (Zhu, 2018).

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 (Das et al., 28 Jan 2026).

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 (Das et al., 28 Jan 2026).

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 (Das et al., 28 Jan 2026). 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 $O(|w||V|^2)$ (Das et al., 28 Jan 2026).

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 (Zhu, 2018).

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 (Zhu, 2018).