Left-to-Right Minima Basis in Combinatorics

• Left-to-right minima basis is a unifying concept that constructs canonical bases by identifying new minimum elements in permutations, lattice vectors, or polynomial monomials.
• It exhibits lexicographic triangularity and filtration compatibility in the symmetric group algebra, facilitating structured eigenvalue computations and efficient algorithmic implementations.
• In lattice theory and interpolation, the greedy construction achieves successive minima and minimal support, offering computational advantages despite inherent limitations in high dimensions.

The left-to-right minima basis ("LRM basis") is a unifying concept in algebraic combinatorics and related fields, arising independently in the theory of group algebras of the symmetric group, the geometry of lattices, and the enumeration of permutation statistics. The LRM basis leverages the combinatorial structure of left-to-right minima in permutations to define canonical, triangular, or extremal bases that reveal deep interactions between representation theory, combinatorics, and computational algebra.

1. Definition and Characterization of Left-to-Right Minima

For a permutation $\sigma=\sigma(1)\ldots\sigma(n)\in S_n$, $\sigma(i)$ is a left-to-right minimum (LR-minimum) if $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$. Equivalently, the statistic $\operatorname{lrmin}(\sigma)$ counts the number of positions $i$ at which $\sigma(i)$ sets a new minimum as one scans the permutation from left to right (Han et al., 2024). In the context of the symmetric group algebra, a “shifted” set $$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$ is also defined, with an associated composition $\mathbf{cLRM}'(w)\vDash n$ (Grinberg et al., 6 Jan 2026).

In lattice theory, “left-to-right minima basis” is synonymous with the “successive minima basis” or “Minkowski basis”: for a lattice $L\subset\mathbb{R}^n$, its successive minima $\lambda_1\leq\lambda_2\leq\ldots\leq\lambda_n$ are defined such that $\sigma(i)$0 is the smallest $\sigma(i)$1 for which there exist $\sigma(i)$2 linearly independent lattice vectors of norm $\sigma(i)$3. The LRM basis is constructed greedily: $\sigma(i)$4 is any shortest nonzero vector, $\sigma(i)$5 is a shortest vector linearly independent from $\sigma(i)$6, guaranteeing $\sigma(i)$7 (Regavim, 2021).

2. LRM Basis in the Group Algebra of the Symmetric Group

Grinberg–Vassilieva (Grinberg et al., 6 Jan 2026) introduced an LRM basis $\sigma(i)$8 in the group algebra $\sigma(i)$9, defined by $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$0, where $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$1 is a Solomon descent algebra element for $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$2. This basis enjoys the following key properties:

• Lexicographic Triangularity: Each $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$3 expands as $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$4 plus a linear combination of lex-order smaller permutations, ensuring that $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$5 forms a basis for $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$6.
• Filtration Compatibility: The subspaces $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$7 coincide with right ideals $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$8 indexed by compositions $\sigma(i)<\min\{\sigma(1),\ldots,\sigma(i-1)\}$9, with respect to the refinement order.
• Triangular Action: The descent algebra acts by (block-)triangular matrices in the LRM basis, leading to effective computations of eigenvalues, e.g., for random walk operators on $\operatorname{lrmin}(\sigma)$0.

The construction is fundamentally connected to noncommutative symmetric functions and Dynkin elements (nested commutators) in the free algebra, which control the inclusion relations between right ideals (Grinberg et al., 6 Jan 2026). This basis is also seen as an analogue of a cellular basis, with potential implications for categorification and generalizations to other Coxeter groups.

3. Successive Minima (LRM) Basis in Lattice Theory

The LRM basis in the context of lattices (Minkowski basis) is defined by sequentially choosing, at step $\operatorname{lrmin}(\sigma)$1, a shortest lattice vector outside the span of previous choices, guaranteeing $\operatorname{lrmin}(\sigma)$2 (Regavim, 2021). This construction has intrinsic significance:

• Minkowski-Reduced Bases: A basis $\operatorname{lrmin}(\sigma)$3 is Minkowski-reduced if $\operatorname{lrmin}(\sigma)$4 is a shortest nonzero vector in $\operatorname{lrmin}(\sigma)$5 and the partial $\operatorname{lrmin}(\sigma)$6-spans are primitive.
• Optimality: The LRM (Minkowski) basis achieves the successive minima, but may not minimize other metrics, e.g., the maximal basis vector norm.

Recent advances (Regavim, 2021) have improved classical bounds on the squared norms of basis vectors: for $\operatorname{lrmin}(\sigma)$7, $\operatorname{lrmin}(\sigma)$8, confirming Schürmann's conjecture and improving Van der Waerden's exponential bound. However, explicit examples show that in high dimension, the last vector in the LRM basis may be exponentially larger than the longest vector in a Korkin–Zolotarev (KZ)-reduced basis, highlighting fundamental limitations of greedy algorithms for basis reduction.

4. LRM Statistics in Permutation Enumeration

The distribution and joint behavior of LRM in permutations, especially in classes such as up–down and down–up (alternating) permutations, are of central combinatorial interest (Han et al., 2024). Relevant generating functions enumerating permutations by number of left-to-right minima are as follows:

• Single Statistic EGF (Up–Down Permutations):

$\operatorname{lrmin}(\sigma)$9

where $i$0 marks the number of LR-minima.

• Combined EGF for Alternating Permutations:

$i$1

with $i$2.

Joint distributions of minima and maxima, as well as mesh-pattern techniques and recursive constructions, enable closed generating functions for these refined statistics. Equidistribution under reverse/complement demonstrates deep symmetry between LR-minima and other extremal statistics across permutation classes.

5. LRM Basis in Polynomial Interpolation

A related, though distinct, concept arises in multivariate polynomial interpolation: the notion of a $i$3-minimal monomial interpolating basis, which is the unique lex-minimal basis with respect to a fixed monomial order (Gong et al., 2020). Here, the LRM principle is formalized via a “reverse reduced basis,” constructed by performing backward Gaussian elimination on the least monomials of a system of linearly independent polynomials, with respect to a chosen term order. The resulting least monomials form a basis with minimal support and triangular evaluation properties for the relevant interpolation conditions.

The algorithmic procedure can be summarized as follows:

• Process polynomials $i$4 by iteratively eliminating the least monomial of each $i$5 from all $i$6.
• The resulting least monomials $i$7 across the reduced polynomials $i$8 constitute the $i$9-minimal interpolating basis.
• This basis guarantees upper-triangularity of the evaluation matrix associated with interpolation, non-singularity, and support minimality.

This reverse reduced approach efficiently bypasses classic Gröbner basis computations for interpolation, leveraging the LRM principle to achieve computational sparsity and structural clarity (Gong et al., 2020).

6. Illustrative Examples and Explicit Computations

Concrete expansions are foundational to understanding the LRM basis. For $\sigma(i)$0, listing permutations, their left-to-right minima, shifted sets, and associated compositions:

Permutation $\sigma(i)$1	$\sigma(i)$2	$\sigma(i)$3	Composition $\sigma(i)$4
123	$\sigma(i)$5	$\sigma(i)$6	(3)
132	$\sigma(i)$7	$\sigma(i)$8	(1,2)
213	$\sigma(i)$9	$$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$0	(1,2)
231	$$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$1	$$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$2	(1,2)
312	$$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$3	$$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$4	(2,1)
321	$$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$5	$$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$6	(1,1,1)

The LRM-basis element $$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$7 for each $$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$8 is constructed as $$\operatorname{LRM}'(w):=\{i-1\,|\,i\in\operatorname{LRM}(w),i>1\}\subset[n-1]$$9, with explicit expansion in the group algebra basis exhibiting lex-triangular structure. The inclusion of these explicit combinatorial mappings is essential for computations in the descent algebra and beyond (Grinberg et al., 6 Jan 2026).

7. Applications, Limitations, and Open Problems

The LRM basis is structurally significant for several domains:

• Representation Theory: The triangularity of the basis under descent algebra action provides generalized eigenvector decompositions and explicit eigenvalues, e.g., for random walks and Hopf algebra connections (Grinberg et al., 6 Jan 2026).
• Basis Reduction in Lattices: While achieving successive minima, the LRM basis may yield poor worst-case maximal norms in high rank, indicating limitations of greedy algorithms and motivating alternative reductions (e.g., KZ reduction) (Regavim, 2021).
• Enumeration and $\mathbf{cLRM}'(w)\vDash n$0-Analogues: The enumeration of permutations by LRM, with closed EGF and joint distributions, enriches the theory of permutation statistics and relates to classic objects such as Euler numbers and Springer numbers (Han et al., 2024).

Open directions include the construction of bases that are simultaneously triangular for both left and right actions of the descent algebra, the extension of the LRM basis to other (e.g., type $\mathbf{cLRM}'(w)\vDash n$1) Coxeter groups, and characterizations of equality cases in lattice reduction extremal problems (Grinberg et al., 6 Jan 2026, Regavim, 2021). In polynomial interpolation, refined algorithmic analyses and connections to sparsity remain active areas of investigation (Gong et al., 2020).