Lehmer Tableaux Clarification

• Lehmer tableaux are misnamed objects that conflate two distinct constructions: Ferrers-diagram Le‐tableaux and permutation encoding Lehmer codes.
• Le‐tableaux are 0/1 fillings of Ferrers diagrams defined by northwest-closed conditions and connected to permutation statistics through bijections.
• Lehmer codes, used in ranking aggregation, are inversion vectors for permutations and are not represented via any tableau formalism.

In the cited arXiv literature, “Lehmer Tableaux” is not a supported technical term. The combinatorial object treated in the Ferrers-diagram setting is Le-tableaux, attributed there to Postnikov, whereas the ranking-theoretic object treated in the aggregation setting is the Lehmer code, also called the inversion vector. Nothing in the cited sources suggests that “Lehmer tableaux” is a standard synonym for Le-tableaux, and no tableau formalism is introduced for Lehmer codes. In this sense, the phrase most plausibly reflects a conflation of two distinct constructions: Le-tableaux on Ferrers diagrams and Lehmer-code encodings of permutations (Selig et al., 2017, Li et al., 2017).

1. Terminological status and scope

Within “EW-tableaux, Le-tableaux, tree-like tableaux and the Abelian sandpile model”, the relevant term is Le-tableaux; the paper explicitly does not use “Lehmer tableaux.” It further states that, in this context, the phrase is not supported terminology and is most likely a mistaken name for Le-tableaux, or possibly a confusion with Lehmer codes / inversion tables, which are different objects (Selig et al., 2017).

The same source notes that Le-tableaux are closely related to, or often identified with, $J$-diagrams in the broader literature, although that synonym is not a major focus there. By contrast, “Efficient Rank Aggregation via Lehmer Codes” is about Lehmer codes as an algorithmic representation of rankings, not about tableaux in the Ferrers-diagram sense. It explicitly discusses vectors, subdiagonal images, and coordinatewise aggregation, but not any object called a Lehmer tableau (Li et al., 2017).

A precise reading of the available evidence therefore separates the terminology as follows. Le-tableaux are $0/1$-fillings of Ferrers diagrams with specific closure conditions. Lehmer codes are subdiagonal integer encodings of permutations. The cited papers do not identify these notions.

2. Definition and internal structure of Le-tableaux

A Le-tableau is defined as a $0/1$-filling of a Ferrers diagram, where a Ferrers diagram is a left-aligned array of boxes with weakly decreasing row lengths from top to bottom. For Le-tableaux, the allowed shapes include a Ferrers diagram with some of its bottommost rows possibly empty. This shape convention is explicitly broader than that used for EW-tableaux (Selig et al., 2017).

The filling is required to satisfy two conditions:

1. Every column has a 1 in some cell.
2. If a cell has a 1 above it in the same column and a 1 to its left in the same row, then it has a 1.

Its size is defined by

$$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$

The same source remarks that this differs from the size convention for EW-tableaux, whose size is one less than rows plus columns, and that this difference is arranged so that both families correspond naturally to permutations of the same length (Selig et al., 2017).

Condition (2) admits an equivalent closure formulation: the set of $1$-cells is northwest-closed with respect to existing $1$s. Equivalently, a cell containing $0$ is forbidden from simultaneously having a $1$ somewhere above it in its column and a $1$ somewhere to its left in its row. This forbidden-configuration view is the formulation used algorithmically in the path constructions appearing later in the paper (Selig et al., 2017).

These rules make Le-tableaux a highly constrained class of Ferrers fillings. Their structure is not described via inversion vectors or Lehmer-type coordinate arrays; rather, it is governed by row–column interaction and by weak-excedance data transported through permutation bijections.

3. Permutation correspondences and shape statistics

A central structural fact is that Le-tableaux are linked to permutations through the standard bijection $\Phi$, under which the rows correspond to weak excedances, that is, indices $0/1$0 with $0/1$1 in the associated permutation. This is the basic permutation-theoretic statistic controlling shape on the Le-tableau side (Selig et al., 2017).

The same paper constructs an explicit bijection from EW-tableaux to Le-tableaux,

$0/1$2

Here $0/1$3 maps EW-tableaux to permutations, $0/1$4 sends descent bottoms to excedance bottoms, $0/1$5 is cyclic right shift

$0/1$6

and $0/1$7 is the inverse map from permutations to Le-tableaux (Selig et al., 2017).

The logic of this construction is entirely statistic-driven. For EW-tableaux, row labels are exactly the descent bottoms of $0/1$8. After applying $0/1$9, these become excedance bottoms. The cyclic right shift then converts excedance bottoms into weak excedance bottoms, which are precisely the row labels on the Le-tableau side. In this manner, shape is transported through a chain of permutation statistics rather than by a direct local rewriting rule (Selig et al., 2017).

The map $0/1$0 itself is defined by an iterative deletion procedure on an EW-tableau: after deleting the entries of the top row, one repeatedly writes down labels of columns with no $0/1$1s from right to left, deletes those columns, then writes down labels of rows with no $0/1$2s from bottom to top, and deletes those rows. The resulting word is a permutation. This mechanism does not involve Lehmer codes or inversion-vector coordinates; it is based on emptiness conditions in rows and columns of a Ferrers filling (Selig et al., 2017).

Enumeratively, the paper quotes a theorem that EW-tableaux of size $0/1$3 with row labels $0/1$4 are equinumerous with $0/1$5-permutations whose excedance bottoms set is $0/1$6. Since EW-tableaux and Le-tableaux are connected bijectively, the effective consequence is that Le-tableaux of a given shape are equinumerous with permutations having the corresponding weak excedance bottoms set (Selig et al., 2017).

4. Direct relations to EW-tableaux, NEW-tableaux, and tree-like tableaux

Beyond the permutation-mediated description, the same paper gives a direct inverse map $0/1$7 from Le-tableaux to EW-tableaux. Starting from a Le-tableau $0/1$8, one forms the EW shape by adding one column to the left border of the Ferrers shape of $0/1$9, of the same length as that border, while preserving any empty bottom rows. The algorithm then uses two boundary labelings and a zig-zag path rule: beginning at an $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$0-label, one moves inward, and whenever a $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$1 is encountered, the path turns west if it was going north and north if it was going west. The exit label determines whether, in the EW-tableau being built, one fills a column with $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$2s or a row with $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$3s, and in what order these operations occur (Selig et al., 2017).

The paper also introduces NEW-tableaux, another $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$4-filling family on Ferrers diagrams, again allowing possible empty bottom rows, defined by the conditions that every column has a $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$5 and that no forbidden rectangle of the EW type occurs. Their bijection to Le-tableaux is

$$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$6

where $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$7 is cyclic down-shift on values: subtract $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$8 from each letter, replacing $$\mathrm{size}(T)=\#\mathrm{rows}+\#\mathrm{columns}.$$9 by $1$0. The source explicitly states that, unlike the EW-to-Le bijection, this NEW-to-Le bijection preserves shape exactly (Selig et al., 2017).

A third family in the same framework is that of tree-tableaux. A tree-tableau is a dotted Ferrers diagram satisfying three conditions: the top-left cell has a dot; every other dotted cell has a dot above it or to its left, but not both; and every row and every column has a dot. The paper recalls a known bijection from tree-tableaux to Le-tableaux and also gives an equivalent local description: after deleting the leftmost column, a cell becomes $1$1 iff it has a dot to its left and a dot weakly above; otherwise it becomes $1$2 (Selig et al., 2017).

Conversely, from a Le-tableau one recovers a tree-tableau by adding a left column with top cell $1$3 and all others $1$4, turning every $1$5 that is highest in its column into a dot, and in each row turning the rightmost restricted $1$6 into a dot, where a restricted $1$7 is a $1$8 with a $1$9 above it in the same column. This bijection is notably local and geometric, in contrast with the statistic-transport mechanism used for the EW–Le correspondence (Selig et al., 2017).

5. Ferrers graphs and the Abelian sandpile model

One of the major themes of the paper is that tableau families on Ferrers diagrams encode graph-theoretic and sandpile-theoretic structures. An EW-tableau on a Ferrers diagram corresponds to an acyclic orientation, with a unique sink, of the associated Ferrers graph. For a cell $1$0, a $1$1 means the edge is oriented $1$2, and a $1$3 means it is oriented $1$4. The forbidden-rectangle condition for EW-tableaux is exactly the condition preventing directed $1$5-cycles, and hence directed cycles, in a Ferrers graph (Selig et al., 2017).

The same paper then defines, from an EW-tableau $1$6, a stable configuration $1$7 on the Ferrers graph $1$8: if $1$9 is a column label, $0$0 is the number of $0$1s in that column; if $0$2 is a row label, $0$3 is the number of $0$4s in that row. Its Theorem 6.1 states that

$0$5

is a bijection from EW-tableaux of shape $0$6 to the minimal recurrent configurations of the Abelian sandpile model on $0$7. Moreover, if $0$8, then toppling $0$9 in the order $1$0 recovers $1$1 (Selig et al., 2017).

Le-tableaux do not directly encode recurrent configurations in that paper. Rather, they acquire this interpretation only indirectly through their bijections with EW-tableaux. A parallel indirect interpretation arises through tree-tableaux: tree-tableaux are identified with spanning trees of the Ferrers graph having external activity $1$2, for any $1$3-compatible edge order, and since tree-tableaux are bijective with Le-tableaux, this supplies another graph-theoretic reading of the Le-tableau family (Selig et al., 2017).

This suggests that the mathematical significance of Le-tableaux in the cited work lies less in any Lehmer-type inversion encoding than in their position inside a web of bijections connecting Ferrers fillings, permutation statistics, spanning trees, acyclic orientations, and minimal recurrent sandpile configurations.

6. Lehmer codes, inversion vectors, and the absence of a tableau formalism

The distinct arXiv paper “Efficient Rank Aggregation via Lehmer Codes” uses Lehmer codes in a wholly different sense. For a permutation $1$4, the Lehmer code is the vector

$1$5

with coordinates

$1$6

The paper explicitly treats Lehmer code and inversion vector as the same object (Li et al., 2017).

Its central interpretive identity is

$1$7

so each coordinate gives the projected insertion position of $1$8 among $1$9. Because the feasible set factors coordinatewise, the paper emphasizes that the coordinates decouple, allowing aggregation of multiple permutations by taking the coordinatewise median or mode of the corresponding Lehmer-code entries and then decoding the resulting vector back to a permutation (Li et al., 2017).

The same source also extends the construction to partial rankings using a companion quantity

$1$0

which captures intervals of possible insertion positions in the presence of ties. The aggregation algorithms are analyzed under Mallows and generalized Mallows models, with explicit sample-complexity bounds for recovering the centroid or a tie-consistent permutation (Li et al., 2017).

None of this constitutes a tableau theory. The paper explicitly states that it does not introduce a triangular array, Ferrers diagram, Young tableau, or any object explicitly called a Lehmer tableau. The representation remains a vector

$1$1

possibly viewed as a subdiagonal image, but not as a tableau-shaped filling (Li et al., 2017).

Accordingly, the phrase “Lehmer Tableaux” has no direct support in either cited source. In the Ferrers-diagram context, the correct object is Le-tableaux. In the ranking-aggregation context, the correct object is the Lehmer code. Any identification of the two is not made in the cited arXiv literature.