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arxiv: 2604.17387 · v1 · submitted 2026-04-19 · 🧮 math.CO

The inversion number statistic for inversion sequences

Lora R. Du , Guo-Niu Han This is my paper

Pith reviewed 2026-05-10 06:05 UTC · model grok-4.3

classification 🧮 math.CO
keywords inversion sequencesinversion numberjoint distributionq-analogComtet formulaStirling numbersMahonian numbersEulerian numbers
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The pith

The inversion number statistic on inversion sequences, studied jointly with four others, recovers the Stirling, Mahonian, Eulerian, Catalan, Narayana, and involution counts via specializations.

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

The paper defines an inversion number for inversion sequences and examines its joint distribution with four additional statistics. This produces a five-parameter generating function whose specializations match several classical enumerative results on permutations. The construction originates from substituting the q-derivative into Comtet's expansion formula. A reader would care because inversion sequences already serve as a basic model in combinatorics, so unifying these distributions under one statistic offers a compact way to see interconnections among known numbers.

Core claim

Inversion sequences, also known as subexcedant sequences, form a fundamental class of objects in enumerative combinatorics. In this paper, we study the joint distribution of five statistics on inversion sequences. While several statistics on inversion sequences have been extensively investigated, our contribution is to introduce the inversion number statistic, originally defined for permutations, into the context of inversion sequences. As special cases, we recover classical permutation statistics, including the Stirling, Mahonian and Eulerian distributions, as well as the Catalan and Narayana numbers. Somewhat unexpectedly, our specializations also include the number of involutions in the 2

What carries the argument

The inversion number statistic on inversion sequences, realized through the q-analog of Comtet's expansion formula obtained by replacing the derivative D with the q-derivative D_q.

If this is right

  • Special cases of the joint generating function yield the q-Stirling numbers of the first kind.
  • Other specializations recover the Mahonian and Eulerian polynomials.
  • Further specializations produce the Catalan and Narayana numbers.
  • One specialization counts the involutions in the symmetric group.

Where Pith is reading between the lines

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

  • The same five-statistic framework might be applied to other combinatorial objects that admit a natural length or height function.
  • The unexpected appearance of involution counts suggests looking for a direct bijection that explains why the statistic combination produces them.
  • One could test whether a similar q-analog construction works for other classical expansion formulas beyond Comtet's.
  • The approach may generate new q-analogs for statistics whose q-versions were previously unknown.

Load-bearing premise

The q-analog substitution into Comtet's expansion formula correctly encodes the joint distribution of the five statistics and its specializations align with the classical definitions of the recovered numbers.

What would settle it

Compute the five-statistic generating function explicitly for inversion sequences of length 4 and verify whether its specializations exactly reproduce the known Narayana numbers and the number of involutions on 4 elements.

Figures

Figures reproduced from arXiv: 2604.17387 by Guo-Niu Han, Lora R. Du.

Figure 1
Figure 1. Figure 1: A geometric representation of the inversion sequence. tel(e) = n − dist(e), uel(e) = n − max(e) − 1, where dist(e) is the number of distinct entries of e. These statistics are, respectively, the number of inversions, the sum of its entries, the number of zero entries, the total number of empty lines (i.e., integers in {0, 1, . . . , n − 1} not present in e), the number of upper empty lines (i.e., integers … view at source ↗
Figure 2
Figure 2. Figure 2: The mapping e ′ → e in Case 1a. (1b) At least one zero in e ′ is mapped to 1, shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The mapping e ′ → e in Case 1b. Case 2: General sequences with 1 ≤ k < n zeros. Let I (k) n be the subset of In with exactly k zeros. For e ′ ∈ I (k) n , each of the n − k non-zero entries is incremented by 1, contributing a factor of p n−k . (2a) All k zeros in e ′ are mapped to 0, see [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The mapping e ′ → e in Case 2a. (2b) At least one zero is mapped to 1, depicted in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The mapping e ′ → e in Case 2b. Collecting all contributions from each case above leads to the recur￾rence relation for Fn+1(x): Fn+1(x) = (z − 1)z n−1 y nx n+1 + p nx  (y − 1)Fn x p  + Tq(Fn(x)) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The lattice path ϕ(e) ∈ D4. While the specialization q = 0 identifies the cardinality of W In with the n-th Catalan number, the refined polynomials reveal deeper struc￾tural symmetries and combinatorial properties, particularly in terms of Dyck paths. Recall that a Dyck path of semilength n consists of up steps (1, 1) and down steps (1, −1) from (0, 0) to (2n, 0), never going below the y-axis. In this cont… view at source ↗
Figure 6
Figure 6. Figure 6: x y 1 2 3 4 0 1 2 3 4 E E N E N N E N y = x [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An involution for lattice path ENEENENN. For the second statement, map each path in Dn to a Dyck path of semilength n by replacing E = (1, 0) with (1, 1) and N = (0, 1) with (1, −1). Under this transformation: • x records the height of the first peak; • z records the height of the last peak. Setting z = x, the exponent records the sum of the heights of the first and the last peak. Therefore, xFn(x; 1, x, 1… view at source ↗
read the original abstract

Inversion sequences, also known as subexcedant sequences, form a fundamental class of objects in enumerative combinatorics. In this paper, we study the joint distribution of five statistics on inversion sequences. While several statistics on inversion sequences have been extensively investigated, our contribution is to introduce the inversion number statistic, originally defined for permutations, into the context of inversion sequences. As special cases, we recover classical permutation statistics, including the Stirling, Mahonian and Eulerian distributions, as well as the Catalan and Narayana numbers. Somewhat unexpectedly, our specializations also include the number of involutions in the symmetric group. Our study arises from a $q$-analog of Comtet's expansion formula obtained by substituting the classical derivative operator $D$ with the $q$-derivative operator $D_q$.

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

1 major / 0 minor

Summary. The paper introduces the inversion number statistic on inversion sequences and studies its joint distribution with four other statistics. It obtains a q-analog of Comtet's expansion formula by substituting the q-derivative D_q for the ordinary derivative D, and shows that specializations of the resulting generating functions recover the Stirling, Mahonian, and Eulerian distributions on permutations, the Catalan and Narayana numbers, and the number of involutions.

Significance. If the substitution step is shown to correctly encode the inversion statistic, the work supplies a unified operator-based framework that recovers multiple classical distributions as special cases of a single joint generating function on inversion sequences. The unexpected appearance of the involution count among the specializations is a notable feature that may point to deeper connections. The approach relies on standard q-analog techniques applied to a cited formula and thereby extends existing enumerative results in a coherent way.

major comments (1)
  1. [Abstract and main derivation] Abstract and main derivation: the central claim that substituting D_q for D in Comtet's expansion formula directly produces the joint distribution of the five statistics (including the newly defined inversion number) is load-bearing for all subsequent specializations. The manuscript must explicitly verify that the action of D_q increments the q-exponent exactly according to the combinatorial definition of inversions on inversion sequences while leaving the other four statistics unchanged; without a concrete computation or small-case check demonstrating this correspondence, the recovery of the Stirling, Mahonian, Eulerian, Catalan, Narayana, and involution counts remains conditional on an unconfirmed operator-statistic alignment.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their thorough reading and insightful comments. The suggestion to provide an explicit verification of the operator-statistic correspondence is well-taken, and we will incorporate it in the revision.

read point-by-point responses

  1. Referee: Abstract and main derivation: the central claim that substituting D_q for D in Comtet's expansion formula directly produces the joint distribution of the five statistics (including the newly defined inversion number) is load-bearing for all subsequent specializations. The manuscript must explicitly verify that the action of D_q increments the q-exponent exactly according to the combinatorial definition of inversions on inversion sequences while leaving the other four statistics unchanged; without a concrete computation or small-case check demonstrating this correspondence, the recovery of the Stirling, Mahonian, Eulerian, Catalan, Narayana, and involution counts remains conditional on an unconfirmed operator-statistic alignment.

    Authors: We acknowledge the importance of explicitly confirming that the substitution of D_q for D encodes the inversion number statistic correctly. Upon review, the manuscript derives the q-analog formally but does not include a direct small-case verification. We will add such a verification in the revised manuscript, computing the action of D_q on the generating function for inversion sequences of small length and matching it to the inversion counts, while confirming the other statistics are preserved. This will solidify the foundation for the subsequent specializations. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external Comtet formula via standard q-substitution to derive new joint distributions

full rationale

The paper begins with Comtet's expansion formula (an external reference) and obtains its q-analog by the standard substitution of the q-derivative operator D_q for D. It then introduces the inversion number statistic on inversion sequences and derives the joint generating function for five statistics. The claimed specializations to Stirling, Mahonian, Eulerian, Catalan, Narayana, and involution numbers function as independent verifications against classically known enumerations rather than as inputs or fitted parameters. No load-bearing step reduces by definition to its own outputs, no self-citation chain is invoked to justify uniqueness or ansatz choices, and the central results remain combinatorially falsifiable outside the paper's own constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard combinatorial definitions of inversion sequences and the listed statistics; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • domain assumption Standard definitions of inversion sequences as subexcedant sequences and of the inversion number, descent, and other statistics as in permutation combinatorics.
    The abstract invokes these objects without re-deriving them, treating them as background from enumerative combinatorics.
  • domain assumption Comtet's expansion formula admits a valid q-analog under substitution of the q-derivative operator.
    The abstract states that the study arises from this substitution, assuming the q-version preserves the necessary algebraic and combinatorial properties.

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