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arxiv: 2211.15277 · v3 · submitted 2022-11-28 · 🧮 math.CO

q-enumeration of type B and D Eulerian polynomials based on parity of descents

Hiranya Kishore Dey , Umesh Shankar , Sivaramakrishnan Sivasubramanian This is my paper

Pith reviewed 2026-05-24 10:54 UTC · model grok-4.3

classification 🧮 math.CO
keywords q-analoguesEulerian polynomialstype Btype Ddescent paritysigned permutationsgenerating functionsinversion number
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The pith

q-analogues enumerate signed permutations in types B and D by tracking parity of descents and ascents together with inversion numbers.

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

Carlitz and Scoville defined a four-variable polynomial that counts ordinary permutations according to the parity of descents and ascents. Pan and Zeng later gave a q-analogue that also incorporates the inversion number and proved a type B version. This paper supplies the corresponding q-analogue for the hyperoctahedral group B_n by using the type B inversion number along with the four parity statistics, and derives an exponential generating function for an analogous five-variable polynomial in type D. The work also produces bivariate q-analogues of Hyatt's recurrences for the Eulerian polynomials of these types and recovers q-analogues of the generating functions for snakes. A reader would care because the results refine the enumeration of Coxeter-group elements with an extra parameter that often interacts usefully with algebraic and combinatorial structures.

Core claim

The paper proves a q-analogue of the type B result of Pan and Zeng by enumerating permutations in B_n with respect to the parity of descents, the parity of ascents, and the type B inversion number. It obtains a q-analogue of the generating function for the type B bivariate alternating descent polynomials. For type D it introduces a five-variable polynomial whose exponential generating function is derived; alternating descents are defined slightly differently from Remmel's earlier version. As by-products the proofs yield bivariate q-analogues of Hyatt's recurrences for the type B and type D Eulerian polynomials, symmetry relations, and q-analogues of the generating functions for snakes of the

What carries the argument

The four- and five-variable generating functions that record the parities of descents and ascents together with the appropriate inversion number; these functions are shown to satisfy recurrences that extend the Pan-Zeng identities.

If this is right

  • Bivariate q-analogues exist for the type B alternating descent polynomials.
  • An exponential generating function is obtained for the five-variable type D polynomial.
  • Bivariate q-analogues of Hyatt's recurrences hold for the type B and type D Eulerian polynomials.
  • Symmetry relations hold for the resulting polynomials.
  • q-analogues exist for the generating functions that count snakes of types B and D.

Where Pith is reading between the lines

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

  • The five-variable treatment required for type D suggests that an extra statistic is needed to capture the structure of even-signed permutations.
  • The slight difference in the definition of alternating descents for type D may produce distinct equidistribution results compared with Remmel's version.
  • When the extra q-parameter is set to 1 the results recover the earlier Pan-Zeng and Carlitz-Scoville identities as special cases.

Load-bearing premise

The definitions of the parity statistics on descents and ascents remain compatible with the type B and type D inversion numbers so that the same recurrence relations used by Pan and Zeng continue to hold.

What would settle it

Direct enumeration of the four-variable polynomial for B_3 or B_4 from the definitions and checking whether the resulting polynomial matches the value obtained from the claimed recurrence.

read the original abstract

Carlitz and Scoville in 1973 considered a four variable polynomial that enumerates permutations in $\mathfrak{S}_n$ with respect to the parity of its descents and ascents. In recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by enumerating permutations with the above four statistice along with the inversion number. Further, they also proved a type B analogue by enumerating signed permutations with respect to the parity of descents and ascents. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by enumerating permutations in $\mathfrak{B}_n$ with the above four statistics and the type B inversion number. We also obtain a $q$-analogue of the generating function for the type B bivariate alternating descent polynomials. We consider a similar five-variable polynomial in the type D Coxeter groups as well and give their egf. Alternating descents for the type D groups were previously also defined by Remmel, but our definition is slightly different. As a by-product of our proofs, we get bivariate $q$-analogues of Hyatt's recurrences for the type B and type D Eulerian polynomials. Further corollaries of our results are some symmetry relations for these polynomials and $q$-analogues of generating functions for snakes of types B and D.

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

0 major / 3 minor

Summary. The manuscript proves q-analogues of the type-B results of Pan and Zeng by enumerating signed permutations in B_n according to four parity-of-descent/ascents statistics together with the type-B inversion number; it also derives a q-analogue of the generating function for the type-B bivariate alternating-descent polynomials. For type D it introduces a five-variable polynomial (with a definition of alternating descents differing slightly from Remmel's) and obtains its exponential generating function. As by-products the paper supplies bivariate q-analogues of Hyatt's recurrences for the type-B and type-D Eulerian polynomials, together with symmetry relations and q-analogues of the generating functions for snakes of types B and D.

Significance. If the claimed identities hold, the work supplies concrete, verifiable extensions of classical q-enumerative results to the hyperoctahedral and even-signed groups, with explicit generating-function recurrences and direct verifications of compatibility with the Pan-Zeng base identities. The bivariate q-analogues of Hyatt's recurrences and the snake generating functions constitute additional reusable combinatorial output.

minor comments (3)
  1. [Introduction] §1 (Introduction): the claim that the type-D definition of alternating descents is 'slightly different' from Remmel's should be accompanied by an explicit side-by-side comparison of the two notions so that readers can immediately see where the statistics diverge.
  2. [Type D section] The reduction to the q=1 case for the type-D five-variable polynomial is asserted but not displayed as a separate corollary; adding a short verification paragraph after the egf derivation would confirm that the new statistics specialize correctly.
  3. [Preliminaries] Notation for the four parity statistics is introduced without a consolidated table; a single table listing the four statistics, their type-B and type-D realizations, and the inversion statistic would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on q-analogues of type B and D Eulerian polynomials and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No circularity; derivations rely on external Pan-Zeng identities and explicit recurrences

full rationale

The paper extends the Pan-Zeng type B result via explicit q-analogues using type B inversion numbers and parity statistics. It supplies generating-function recurrences, inductive arguments, and bivariate q-analogues of Hyatt's recurrences that are verified directly under the chosen statistics. These steps are independent of any self-citation chain or fitted-parameter renaming; the cited Pan-Zeng work is external and the constructions are shown compatible by direct recurrence verification. No load-bearing step reduces by the paper's own equations to a definition or fit internal to this manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard generating-function identities for q-analogues and on the compatibility of the chosen descent and inversion statistics with the recurrence relations of Pan and Zeng. No new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption The type B and type D inversion numbers and descent statistics satisfy the same recurrence relations as their classical counterparts when q=1.
    Invoked when the authors state that their q-analogues specialize to the known Pan-Zeng and Carlitz-Scoville results.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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