Lattice of Weak Compositions

• The lattice of weak compositions is a combinatorial structure formed by ordered s-tuples of nonnegative integers summing to n, organized via dominance order.
• It features explicit meet and join operations derived from prefix-sum sequences, enabling precise computation of grading, Möbius functions, and poset invariants.
• This structure has practical applications in coding theory, enumerative combinatorics, and geometric inequalities, notably linking to optimal Lee-metric anticodes and generalized FKG inequalities.

A lattice of weak compositions is a combinatorial structure arising from the set of ordered $s$-tuples of nonnegative integers summing to a fixed integer $n$, equipped with the dominance (majorization) order. This framework naturally encodes a rich distributive lattice structure, with direct applications to coding theory (notably to optimal Lee-metric anticodes over chain rings), enumerative combinatorics, and mixed geometric inequalities. The precise lattice operations, grading, Möbius function, and associated poset-invariants admit explicit formulas and detailed structural understanding, enabling significant generalizations of classical inequalities and deep links to linear algebraic and coding-theoretic objects.

1. Definition and Dominance Order

Fix nonnegative integers $n$ (the total weight) and $s \ge 1$ (the length of each composition). The set of weak $s$-compositions of $n$ is

$$\Delta_s(n) = \left\{ \alpha = (\alpha_1, \dots, \alpha_s) \in \mathbb{Z}_{\ge 0}^s \mid \sum_{i=1}^s \alpha_i = n \right\}.$$

The partial order is given by dominance: $$\alpha \succeq \beta \quad \Longleftrightarrow \quad \sum_{i=1}^k \alpha_i \ge \sum_{i=1}^k \beta_i \quad \text{for all } k=1,\dots,s.$$ This order can be succinctly captured via prefix-sum sequences $\widehat\alpha_k = \sum_{i=1}^k \alpha_i$; then $\alpha \succeq \beta$ if and only if $\widehat\alpha_k \ge \widehat\beta_k$ for all $k$ (Bariffi et al., 12 Jan 2026, Kerner et al., 2014).

Symmetry under coordinate permutations is governed by the symmetric group $\mathfrak{S}_s$, important for passing between compositions and partitions but not essential in basic lattice behavior.

2. Lattice Structure and Explicit Meet/Join

The poset $(\Delta_s(n), \succeq)$ is a finite distributive lattice. The componentwise maximum and minimum in the prefix-sum representation yield the join and meet: $$\widehat{\alpha \vee \beta} = \big( \max(\widehat\alpha_1,\widehat\beta_1), \dots, \max(\widehat\alpha_s,\widehat\beta_s) \big)$$

$$\widehat{\alpha \wedge \beta} = \big( \min(\widehat\alpha_1,\widehat\beta_1), \dots, \min(\widehat\alpha_s,\widehat\beta_s) \big)$$

The original coordinates are recovered by difference: $$(\alpha\vee\beta)_1 = \max(\widehat\alpha_1,\widehat\beta_1),\quad (\alpha\vee\beta)_k = \max(\widehat\alpha_k,\widehat\beta_k) - \max(\widehat\alpha_{k-1},\widehat\beta_{k-1})\ (k\ge2)$$ and similarly for the meet operation (Bariffi et al., 12 Jan 2026).

The minimum (bottom) element is $\mathbf{0} = (0, \ldots, 0, n)$, and the maximum (top) element is $\mathbf{1} = (n, 0, \ldots, 0)$. Distributivity stems from the distributivity of $\min$ and $\max$ in the prefix-sum domain.

3. Grading, Covering Relations, and Möbius Function

Grading and Ranks

Every saturated chain from $\mathbf{0}$ to $\mathbf{1}$ has length $sn$, corresponding combinatorially to the process of moving $n$ units from the last coordinate to the first, one unit and one position at a time. The rank function is

$\rk(\alpha) = \sum_{k=1}^s (s-k)\,\alpha_k = \sum_{k=1}^s \widehat\alpha_k - \sum_{k=1}^s k \alpha_k$

and increments by 1 along cover relations.

Covering Relations

A covering step $\alpha \prec \beta$ occurs precisely if $\beta$ is obtained from $\alpha$ by moving a single unit from a coordinate $j+1$ to $j$: $$\beta_j = \alpha_j + 1, \ \beta_{j+1} = \alpha_{j+1} - 1, \ \beta_i = \alpha_i\ \forall i \ne j, j+1$$ This operation encodes a local "unit transfer" along adjacent positions.

Boolean Sublattices and Möbius Function

Given $\alpha\in\Delta_s(n)$, the subset of all compositions obtainable by any subset of allowed unit-moves yields a Boolean sublattice, with dimension given by the Hamming weight of the tail $(\alpha_2, \ldots, \alpha_s)$ (Bariffi et al., 12 Jan 2026).

The Möbius function on intervals takes the explicit form

$$\mu(\alpha,\beta) = \begin{cases} (-1)^{\sum_i (\beta_i-\alpha_i)} & \text{if } \beta \text{ belongs to the Boolean sublattice from } \alpha,\ 0 & \text{otherwise.} \end{cases}$$

Enumeration

The cardinality is given by the stars-and-bars formula: $|\Delta_s(n)| = \binom{n+s-1}{s-1}$ Rank-generating polynomials and finer enumerative invariants are available via standard poset techniques, though closed formulas for chain counts at a given rank are not generally explicit (Bariffi et al., 12 Jan 2026, Kerner et al., 2014).

4. Anti-Isomorphism and Symmetric Group Actions

The involution $\alpha \mapsto (\alpha_s, \ldots, \alpha_1)$ reverses the dominance order, exhibiting an anti-isomorphism within the lattice. Action of $\mathfrak{S}_s$ by permutation of coordinates permutes the structure among different orbits, allowing passage between labeled compositions and unlabeled integer partitions (Kerner et al., 2014).

While $(\Delta_s(n),\succeq)$ is a distributive lattice, the quotient by $\mathfrak{S}_s$ (partitions with at most $s$ parts) yields the classical partition lattice ordered by dominance, which is not distributive but retains meet and join operations via the same partial-sum constructions.

5. Correspondence to Optimal Lee-Metric Anticodes

A key application is the bijection between the lattice of weak compositions and the inclusion-ordered lattice of optimal Lee-metric anticodes over the chain ring $\mathbb{Z}/p^s\mathbb{Z}$ (with $p\ne2$). Each anticode's support subtype $a = (a_0,a_1, \ldots, a_s) \in \Delta_{s+1}(n)$ encodes the counts of coordinates generating the ideal $(p^i)$. Explicitly,

$$a_i = \left|\{\text{coordinates generating the ideal } (p^i)\}\right|, \quad \sum_{i=0}^s a_i = n$$

The canonical generator matrix is block-diagonal with $I_{a_0}, p\,I_{a_1}, \ldots, p^{s-1} I_{a_{s-1}}$, and full degeneracy in the last $a_s$ coordinates.

Inclusion of anticodes corresponds exactly to dominance: $\mathcal{A}_a\subseteq\mathcal{A}_b$ if and only if $a \succeq b$. This establishes a poset-isomorphism: $$\Delta_{s+1}(n) \longleftrightarrow \{\text{optimal Lee-metric anticodes in } (\mathbb{Z}/p^s\mathbb{Z})^n\}$$ providing combinatorial and algebraic invariants for the study of error-correcting codes (Bariffi et al., 12 Jan 2026).

6. Generalized FKG Inequality and Geometric Applications

The lattice of weak compositions underlies a generalized Fortuin-Kasteleyn-Ginibre (FKG) correlation inequality for functions on $K_{n,r}$, as established by Kerner–Némethi (Kerner et al., 2014). For non-negative, non-decreasing (in the dominance order) functions $f,g$ on $K_{n,r}$, with $f$ symmetric,

$$\left(\frac{1}{|K_{n,r}|} \sum_{k} f(k)\right) \left(\frac{1}{|K_{n,r}|} \sum_{k} g(k)\right) \le \frac{1}{|K_{n,r}|} \sum_{k} f(k) g(k)$$

Equality characterizations and dual inequalities for non-increasing $g$ complete the statement. The proof exploits stratification by the number of zeros and a Chebyshev-type summation argument.

This result generalizes mixed volume inequalities such as Aleksandrov–Fenchel and Teissier's mixed covolume inequalities, with the weak composition lattice providing the underlying combinatorial structure for these geometric inequalities.

7. Examples and Explicit Computations

For $s=4$, $n=3$, $\Delta_4(3)$ consists of all ordered 4-tuples of non-negative integers summing to 3. The Hasse diagram arranges these into four layers by rank; each cover operation corresponds to a local left-move of a unit. In coding theory, each weak composition in $\Delta_3(3)$ corresponds to a unique class of optimal Lee-metric anticodes in $(\mathbb{Z}/p^2\mathbb{Z})^3$, with inclusion relationships recovering the dominance structure.

The table below summarizes the correspondence for $s+1=3,\,n=3$:

Weak composition $a$	Support subtype	Generator matrix (up to perm.)
$(2,1,0)$	$(2,1,0)$	$\diag(I_2, p\,I_1)$
$(1,1,1)$	$(1,1,1)$	$\diag(I_1, p\,I_1,p^2\,I_1)$

Dominance, e.g., $(2,1,0)\succeq(1,1,1)$, matches precisely with anticode inclusion.

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In summary, the lattice of weak compositions with dominance order is a fundamental structure in algebraic combinatorics, encoding distributive, graded lattices, supporting Boolean sublattices, with explicit Möbius function and enumerative data, and provides powerful correspondences with inclusion orders of special error-correcting code families, generalized correlation inequalities, and geometric volume inequalities (Bariffi et al., 12 Jan 2026, Kerner et al., 2014).