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arxiv: 2606.21532 · v1 · pith:BI3E6KR7new · submitted 2026-06-19 · 🧮 math.CO

Relating tournaments and permutations with xrays

Matthew Davis , Michael W. Schroeder This is my paper

Pith reviewed 2026-06-26 13:42 UTC · model grok-4.3

classification 🧮 math.CO
keywords tournamentspermutationsxraysscore sequences2-tournamentsbinary xrayscombinatorial constructions
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The pith

A construction uses any permutation to build 1- and 2-tournaments whose score sequences equal the permutation's xray.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines transitive tournament decompositions of k-tournaments and then supplies an explicit construction that turns a given permutation into both 1-tournaments and 2-tournaments. The score sequences of the resulting directed graphs match the xray values of the permutation exactly as described in prior work. This construction makes the conjectures of Bebeacua et al. for binary xrays and of Brualdi and Fritscher for general xrays into special cases of one broader statement about xrays whose terms are restricted to any admissible set. The authors supply supporting evidence for the general statement and examine the binary and other restricted cases in detail.

Core claim

We first introduce the concept of a transitive tournament decomposition of k-tournaments, then present a construction by which a permutation is used to build 1- and 2-tournaments whose score sequences agree with the xray of the permutation in the manner outlined by Bebeacua et al. and Brualdi and Fritscher. We close with an investigation of xrays with restricted terms, including binary xrays, and show that the recent conjectures by Bebeacua et al. and Brualdi and Fritscher are special cases of a more general statement, which we conjecture and for which we provide supporting evidence.

What carries the argument

The transitive tournament decomposition of k-tournaments, which is used to turn the xray entries of a permutation into the out-degrees of the constructed 1- and 2-tournaments.

If this is right

  • The conjectures of Bebeacua et al. and Brualdi and Fritscher hold as immediate special cases.
  • The same correspondence holds for every admissible restriction on the terms of the xray.
  • Every admissible xray term set is realized as the score sequence of some tournament obtained from the construction.
  • Binary xrays are realized precisely by the score sequences of ordinary tournaments via this method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit mapping supplies a uniform generator for families of tournaments and 2-tournaments indexed by permutations.
  • The method suggests a route for extending the correspondence to k-tournaments when k exceeds 2 by iterating the decomposition.
  • Verification of the general conjecture on small permutations would give concrete evidence that can be checked by direct computation of xrays and constructed scores.

Load-bearing premise

The construction always produces a valid tournament or 2-tournament whose scores exactly equal the given xray for every permutation and every admissible choice of restricted terms.

What would settle it

Any specific permutation together with an admissible xray term set for which the constructed directed graph either contains a pair of vertices with no edge or two edges in opposite directions, or for which at least one vertex score differs from the corresponding xray value.

Figures

Figures reproduced from arXiv: 2606.21532 by Matthew Davis, Michael W. Schroeder.

Figure 1
Figure 1. Figure 1: (a) A transitive tournament and its adjacency matrix. (b) A non-transitive tournament and its adjacency matrix. 0 1 3 2 0 1 1 2 1 0 1 2 1 1 0 1 0 0 1 0 0 1 3 2 0 1 0 1 0 0 0 1 1 1 0 1 0 0 0 0 0 1 3 2 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 0 (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A 2-tournament with its adjacency matrix (a), and two transitive tourna￾ments with their adjacency matrices (b and c) which comprise a transitive tournament decomposition of the 2-tournament. Let AT be the adjacency matrix of T, where the rows and columns of AT are indexed by Zn. Observe that AT + A′ T = Jn − In, where A′ T is the transpose of AT , In is the identity matrix of order n, and Jn is the n × n … view at source ↗
Figure 3
Figure 3. Figure 3: The transitive tournaments T(2310) (a) and T(0123) (b) with their adjacency matrices, and their sum T 2 (2310) (c), which is T(2310) + T(0123). permutations and transitive tournaments, and we denote ϕT as the permutation corresponding to T in this way. Note that ϕT (i) = sT (i) for each i ∈ Zn. With this in mind, we can also find a 2-tournament that naturally corresponds to a permutation with respect to it… view at source ↗
Figure 4
Figure 4. Figure 4: Producing a Skolem sequence from a permutation with a binary xray. xrays, which in our context, is equivalent to the following. Let ϕ ∈ Sn and suppose c(ϕ) ∈ Cn. Construct a matrix by replacing each 1 in Pϕ with the xray index of the 1. Project the nonzero entries of this matrix up and to the left. Concatenating these projections, then adding 1 to each term produces a Skolem sequence. For example, let ϕ = … view at source ↗
Figure 5
Figure 5. Figure 5: A 2-tournament with score sequence (3, 3, 3, 7, 7, 7). c(ϕ) = (n − 1 : n) = s(2T), but as discussed earlier, since T is not transitive, 2T does not have a TTD. We have thus established that, given a score sequence s ∈ S2 n , there may be tournaments T ∈ T 2 n for which s(T) = s and T does not have a TTD. This leads to the following question: Question 1. Let n be a positive integer and s ∈ S2 n . Does there… view at source ↗
Figure 6
Figure 6. Figure 6: The 2-tournament (a) constructed in Example 3.6, and two transitive tournaments T(1023) (b) and T(3102) (c) which comprise a TTD of the 2-tournament. Note that the latter condition relating T to T ′ is analogous to embedding a partial ordering of Zn in a total ordering. For a tournament W ∈ Tn, let (Zn, ≺W ) be the partial ordering of Zn such that for each x, y ∈ Zn, x ≺W y if and only if sW (x) < sW (y). … view at source ↗
Figure 7
Figure 7. Figure 7: The two nonisomorphic tournaments with score sequence (1, 1, 2, 3, 3). The edges in bold highlight the edges necessary to be reversed to yield the transitive tournament T01234. 0 >T1 4. Let T ′ ∈ T5 be a transitive tournament such that, if sT1 (x) < sT1 (y), then x <T′ y. Then T ′ = T(ϕ) for some ϕ ∈ {01234, 10234, 01243, 10243}. Suppose that T1 + T ′ = R1 + R2 for transitive R1, R2 ∈ T5, and without loss … view at source ↗
Figure 8
Figure 8. Figure 8: Adjacency matrices for tournaments with score sequences in S 2 (24 ) and S 4 (14 ), respectively, which follow the correspondence outlined in Example 3.14. We may loosely interpret T ′ as a 2k-tournament in the following way. The vertices of T ′ correspond to teams of players from T with the same score in T, and ωT′ (i, j) is the average number of wins each player in team i has against the players in team … view at source ↗
Figure 9
Figure 9. Figure 9: The adjacency matrices for the digraph T ′ when (a) ϕ = 21634705 or 61034725, and when (b) ϕ = 25034761 or 27034165, as outlined in Example 3.14. such that T 2 (ϕ) = T. There are four permutations ϕ for which c(ϕ) = s, namely ϕ ∈ {21634705, 25034761, 61034725, 27034165}. However, in each case, the resulting digraph T ′ on Z4 with outdegrees {1, 5, 7, 11} does not have integral edge weights; see [PITH_FULL… view at source ↗
read the original abstract

In a 2005 paper, Bebeacua et al. investigated the xrays of permutations, and conjectured a correspondence between binary xrays and score sequences of tournaments. In 2014, Brualdi and Fritscher conjectured a possible correspondence between score sequences of $2$-tournaments and (not necessarily binary) xrays of permutations. In this paper, we first introduce the concept of a transitive tournament decomposition of $k$-tournaments, then present a construction by which a permutation is used to build $1$- and $2$-tournaments whose score sequences agree with the xray of the permutation in the manner outlined by Bebeacua et al. and Brualdi and Fritscher. We close with an investigation of xrays with restricted terms, including binary xrays, and show that the recent conjectures by Bebeacua et al. and Brualdi and Fritscher are special cases of a more general statement, which we conjecture and for which we provide supporting evidence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces the notion of a transitive tournament decomposition for k-tournaments and gives an explicit construction that, for any permutation, produces a 1-tournament and a 2-tournament whose score sequences equal the x-ray of the permutation (in the sense of Bebeacua et al. and Brualdi-Fritscher). It then shows that the two existing conjectures are special cases of a more general statement about x-rays with restricted terms, states that general statement as a conjecture, and supplies supporting evidence (primarily for binary x-rays).

Significance. If the construction is shown to be valid for every permutation and every admissible term restriction, the work would resolve the two cited conjectures and supply a uniform combinatorial bridge between permutation x-rays and tournament score sequences. The explicit construction and the reduction of prior conjectures to a single statement are the main contributions.

major comments (3)
  1. [§3] §3 (Construction): the manuscript asserts that the mapping from permutation to 1-tournament always yields a complete oriented graph (exactly one directed edge between every pair), yet supplies neither an inductive argument nor a direct verification that out-degrees are correctly realized for arbitrary n. A small-n exhaustive check or a short proof that the orientation rule never produces bidirectional or missing edges is required.
  2. [§4] §4 (2-tournaments and general x-rays): the claim that the same construction produces a 2-tournament whose score sequence exactly matches an arbitrary admissible x-ray term set rests on the definition of the edge multiplicities; no general argument is given that the resulting multiset of out-degrees coincides with the prescribed x-ray for every choice of term restrictions. The supporting evidence appears to be computational for small cases only.
  3. [§5] §5 (Restricted-term conjecture): the new general conjecture is stated, but the reduction showing that the Bebeacua et al. and Brualdi-Fritscher statements are literal special cases is only sketched; an explicit substitution of the term restrictions into the general statement is needed to confirm the claim.
minor comments (2)
  1. [§2] Notation for x-ray terms is introduced without a displayed equation; a single displayed definition would improve readability.
  2. Several figures illustrating the construction would benefit from explicit labeling of the permutation and the resulting score sequence.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We respond to each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (Construction): the manuscript asserts that the mapping from permutation to 1-tournament always yields a complete oriented graph (exactly one directed edge between every pair), yet supplies neither an inductive argument nor a direct verification that out-degrees are correctly realized for arbitrary n. A small-n exhaustive check or a short proof that the orientation rule never produces bidirectional or missing edges is required.

    Authors: We agree that the manuscript lacks an explicit verification. In the revision we will add a short inductive proof on n establishing that the orientation rule produces exactly one directed edge between each pair and realizes the correct out-degrees. We will also include an exhaustive computational check for all permutations of length at most 6. revision: yes

  2. Referee: [§4] §4 (2-tournaments and general x-rays): the claim that the same construction produces a 2-tournament whose score sequence exactly matches an arbitrary admissible x-ray term set rests on the definition of the edge multiplicities; no general argument is given that the resulting multiset of out-degrees coincides with the prescribed x-ray for every choice of term restrictions. The supporting evidence appears to be computational for small cases only.

    Authors: The observation is accurate: a general argument is not supplied. We will insert a direct proof that, for any admissible term restriction, the out-degree multiset of the constructed 2-tournament equals the prescribed x-ray, using the multiplicity definition. This will complement the existing small-case computations. revision: yes

  3. Referee: [§5] §5 (Restricted-term conjecture): the new general conjecture is stated, but the reduction showing that the Bebeacua et al. and Brualdi-Fritscher statements are literal special cases is only sketched; an explicit substitution of the term restrictions into the general statement is needed to confirm the claim.

    Authors: We accept that the reduction requires explicit verification. The revised manuscript will contain the direct substitution of the term restrictions from each prior conjecture into the general statement, confirming they arise as special cases. revision: yes

Circularity Check

0 steps flagged

Explicit construction from permutations to tournaments is self-contained

full rationale

The paper introduces the independent concept of transitive tournament decomposition and then defines an explicit construction that maps any permutation to 1- and 2-tournaments whose score sequences are asserted to match the permutation's xray. No parameters are fitted to data, no predictions are made from subsets of the same data, and no load-bearing steps rely on self-citation of the authors' prior results. The cited conjectures of Bebeacua et al. and Brualdi-Fritscher are external and treated as special cases of a new conjecture; the construction itself supplies the claimed correspondence without reducing to a definitional tautology or renamed input.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no equations or definitions, so no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.1-grok · 5700 in / 1235 out tokens · 15479 ms · 2026-06-26T13:42:49.743825+00:00 · methodology

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