A note on the distribution of major index for Schr\"oder paths
Pith reviewed 2026-05-25 19:03 UTC · model grok-4.3
The pith
Two formulas for the major index distribution on Schröder paths hold in every case after the 1993 proof is completed.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
After an error in the original argument is corrected, the two formulas for the distribution of the major index on Schröder paths are shown to hold for all possible relations among the numbers of east, down and north steps.
What carries the argument
The major index on a Schröder path, defined as the sum of the positions of its descents, together with the two explicit formulas that count paths by this statistic.
If this is right
- The q-generating function for the major index on Schröder paths is known explicitly for every choice of step-type counts.
- The distribution can be read off uniformly whether the number of east steps is smaller than, equal to, or larger than the number of down steps.
- Any enumeration or moment that depends on these distributions is now available without case distinctions.
Where Pith is reading between the lines
- The closed formulas allow direct calculation of the mean and variance of the major index for Schröder paths of any length.
- The same verification technique could be applied to other descent-like statistics on the same paths.
- A bijective proof of the formulas might now be sought once the algebraic identity is settled.
Load-bearing premise
The two formulas stated in 1993 are the correct expressions for the distribution, and the only defect was an incomplete proof.
What would settle it
Direct enumeration of all Schröder paths of small length, computation of their major indices, and comparison of the resulting counts against the numbers predicted by the two formulas.
read the original abstract
Bonin, Shapiro and Simion (1993) gave two formulas on the distribution of major index for Schr\"oder paths, and proved their result for the case $E<D<N$. In this short note, we correct an error in their proof, and give a complete proof for all cases.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript identifies an error in the 1993 proof by Bonin, Shapiro and Simion of two formulas for the distribution of the major index on Schröder paths (specifically for the case E < D < N) and supplies a complete, self-contained proof that covers all parameter regimes.
Significance. If the supplied proof is correct, the note completes a 1993 result on the q-enumeration of Schröder paths by major index. The explicit correction and uniform treatment of all cases removes the previous gap and may serve as a reference for further work on statistics of lattice paths and Dyck-like objects.
minor comments (1)
- [Abstract and §1] The abstract and introduction should explicitly restate the two 1993 formulas (including the precise definitions of the parameters E, D, N) so that the corrected proof can be read without immediate recourse to the 1993 paper.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary accurately reflects the contribution of the note.
Circularity Check
No significant circularity
full rationale
The paper is a short correction note whose sole purpose is to supply a complete proof of two formulas originally stated in Bonin-Shapiro-Simion 1993. The 1993 statements are treated strictly as external targets to be proved; the note does not derive any new quantities from its own fitted parameters, does not invoke self-citations for load-bearing steps, and contains no ansatz or uniqueness claim that reduces to prior work by the same author. The derivation chain is therefore self-contained against the external benchmark of the 1993 formulas.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Definitions of Schröder paths, the major index statistic, and the two distribution formulas as stated in Bonin, Shapiro and Simion (1993).
Reference graph
Works this paper leans on
-
[1]
Bonin J, Shapiro L, Simion R. Some q-analogues of the Schr¨ oder n um- bers arising from combinatorial statistics on lattice paths[J]. Journ al of Statistical Planning and Inference, 1993, 34(1): 35-55
work page 1993
-
[2]
F¨ urlinger J, Hofbauer J. q-Catalan numbers[J]. Journal of Combinatorial Theory, Series A, 1985, 40(2): 248-264
work page 1985
-
[3]
MacMahon P A. Combinatory analysis(Vol. 2)[M]. Cambridge Univer- sity Press, Cambridge, 1918. Reprinted by Chelsea, New York, 196 0. Address: School of Mathematics and Computational Science, Hunan Univer - sity of Science and Technology, Xiangtan 411201, China. E-mail address : xmchen@hnust.edu.cn 5
work page 1918
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.