Recursions for rational q,t-Catalan numbers

Eugene Gorsky; Mikhail Mazin; Monica Vazirani

REVIEW 3 major objections 6 minor 21 references

Recursions for rational q,t-Catalan numbers

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that all rational q,t-Catalan power series, coprime or not, are computed by a single fixed-length binary recursion with a unique solution, and connects the result to torus-link Khovanov-Rozansky homology at a=0.

desk verdict The fixed-length binary recursion for rational q,t-Catalan series is a real, self-contained combinatorial result; the advertised Khovanov–Rozansky verification rests on an unpublished theorem and should be treated as conditional. read the letter →

arxiv 1908.11763 v1 pith:T5GXLDQ7 submitted 2019-08-30 math.CO

Eugene Gorsky , Mikhail Mazin , Monica Vazirani This is my paper · ORCID

classification math.CO MSC 05A3005A1705E10

keywords rationalqt-CatalannumbersbinarysequencerecursionsimultaneouscorepartitionsinvariantsubsetstoruslinkhomologyKhovanov-RozanskyDyckpathsSchröderpowerseries

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a recursion, indexed by binary sequences of fixed length $M+N$, that computes the rational $q,t$-Catalan power series $c_{M,N}(q,t)$ for every pair of positive integers $(M,N)$. The recursion has a unique solution once the all-ones sequence is declared to be $1$, and it works in the non-coprime case where $c_{M,N}$ becomes a rational function rather than a polynomial. The same series is then connected, at the specialization $a=0$, to the Poincaré series of the Khovanov-Rozansky homology of the $(M,N)$ torus link, reproducing and extending known recursions from link homology. A reader should care because the paper reduces a central object of rational Catalan combinatorics to a simple deterministic procedure and gives explicit formulas in small cases.

What carries the argument

The central mechanism is the shift map $\rho$ on invariant subsets: remove $0$ if it lies in the subset and then subtract $1$ from every remaining element. Under this shift, area changes by either $0$ or $1$ and codinv changes by a controlled correction measured by $\lambda(u)$, the number of $N$-generators in $[N,N+M-1]$; the three cases of the recursion correspond to whether $0$, $M$, and $N$ are gaps or occupied. This turns the infinite sum defining $P_u$ into a finite decision tree whose cycles are geometric series in $q$, which is what makes the computation effective.

What would settle it

Compute the $a=0$ Khovanov-Rozansky Poincaré series of the $(2,2)$ Hopf link directly from the chain complex and compare it with $t^{-\delta(2,2)}(1-q)^{-1}c_{2,2}=(q+t-qt)/(1-q)$; any mismatch would disprove the topological identification. Independently, direct enumeration of $(2,2)$- or $(4,6)$-invariant subsets can be compared with the recursion's output to test the purely combinatorial claim.

Watch

Extended reading notes

Core claim

For an admissible binary sequence $u=(u_0,\ldots,u_{M+N-1})$, let $P_u(q,t)$ count $(M,N)$-invariant subsets whose intersection with $[0,M+N-1]$ is recorded by $u$, weighted by $q^{\mathrm{area}}t^{\mathrm{codinv}}$. The paper proves that the shift map on invariant subsets gives the three-case recursion $P_u=q(P_v+P_{v'})$ when $u_0=u_M=u_N=0$; $P_u=qP_v$ when $u_0=0$ and at least one of $u_M,u_N$ is $1$; and $P_u=t^{\lambda(u)}P_v$ when $u_0=u_M=u_N=1$, where $v=(u_1,\ldots,u_{M+N-1},1)$, $v'=(u_1,\ldots,u_{M+N-1},0)$, and $\lambda(u)$ counts $N$-generators in the interval $[N,N+M-1]$. With the initial condition $P_{1^{M+N}}=1$ the recursion has a unique solution, so every $P_u$ is a rational function whose denominator is a product of factors $1-q^{\ell}$. A normalization identity then shows that $(1-q)c_{M,N}(q,t)$ is recovered from $P_{0^{M+N}}$, and comparison with the link-homology recursion gives $R_{0^M,0^N}(q,t,0)=t^{-\delta(N,M)}(1-q)^{-1}c_{M,N}(q,t)$.

Load-bearing premise

The load-bearing premise is the cited theorem from [17] identifying $R_{0^M,0^N}(q,t,a)$ with the Poincaré series of the $(M,N)$ torus link; the paper's own recursion is self-contained, but the Khovanov-Rozansky connection loses its verification if that theorem is unavailable or wrong.

Editorial extensions

If this is right

For every positive $M,N$, $c_{M,N}(q,t)$ is a rational function with denominator $(1-q)^{d-1}$ with $d=\gcd(M,N)$, and the recursion computes it using only finitely many $q$-geometric cycles.
The $a=0$ Poincaré series of the Khovanov-Rozansky homology of the $(M,N)$ torus link equals $t^{-\delta(N,M)}(1-q)^{-1}c_{M,N}(q,t)$, matching the known conjectures for all positive $M,N$.
The colored extension expresses the Poincaré series of the $\mathrm{Sym}^d$-colored $(m,n)$ torus knot as $\prod_{i=1}^d(1-qt^{i-d})^{-1}R_{0^{M-d1^d},0^{N-d1^d}}(q,t,a)$, giving a combinatorial interpretation at $a=0$ in terms of the polynomials $P_{0^{M+N-d1^d}}$.
Adding a variable $a$ that records double cogenerators extends the recursion to rational $q,t$-Schröder power series, matching the full three-variable link-homology recursion.
The denominator of the link-homology series $R_{x,y}(q,t,a)$ is a power of $(1-q)$, and the same reduction gives the stated denominator for $P_{0^{M+N}}$ and $c_{M,N}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The recursion is short enough that it could serve as a definition of the rational $q,t$-Catalan series in formal verification settings, independent of Dyck paths or cores.
Because the recursion is symmetric in $M$ and $N$ up to the normalization, it may yield a direct combinatorial proof of $q,t$-symmetry of $(1-q)^{d-1}c_{M,N}$ in the non-coprime case without invoking the shuffle conjecture.
One can automate the decision trees to produce closed formulas for infinite families such as $(d,d)$ or $(2d,3d)$, possibly exposing patterns in the rational Catalan coefficients.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The one thing you should know: Theorem 2.14 is the real contribution. It gives a recursion labeled by binary sequences of length M+N for the rational q,t-Catalan power series, both in the coprime and non-coprime cases, with a uniqueness theorem and explicit denominator control. The recursion is new—earlier recursions by Elias, Hogancamp, and Mellit used pairs of words of varying length. The reformulation in Section 4, showing that the binary recursion is equivalent to the Hogancamp–Mellit R-recursion after forgetting bullets, is a clean synthesis and I found it convincing.

What is good: the combinatorial core is self-contained. The shift map and the three-case recursion are stated precisely, and the proof of Theorem 2.14, while terse, is credible. Property (d) in the proof is the only place where I wanted a few more words, but the claimed change in codinv is exactly what the λ statistic measures, so the step checks out. The uniqueness argument in Theorem 2.19 uses a cycle argument with a 'not hard to see' step; I was initially alarmed, but the periodicity lemma (Lemma 2.18) supplies what is needed. The examples for (2,2), (3,3), and (4,6) are worked in detail, and the q,t-symmetry checks on (1-q)^{d-1} c_{M,N} are a good sanity test.

The soft spot is the topological half. Corollary 4.14 equates R_{0^M,0^N}(q,t,0) with t^{-δ}(1-q)^{-1} c_{M,N}(q,t), and Section 4.2 identifies R with the Poincare series of the torus link via Theorem 4.16, cited as '[17] In preparation.' That theorem is not stated or proved here, and the same reference supplies the well-definedness of the R-recursion used in Theorem 4.13. So the advertised verification of the Khovanov–Rozansky connection is conditional on unpublished work. This is a missing-support problem, not an internal inconsistency. If [17] lands and the theorem is as stated, the topological story goes through.

Who this is for: algebraic combinatorialists working on rational Catalan combinatorics, and people in link homology who want a fixed-length recursion to compute with. The paper deserves a serious referee. I would send it out, with a request that the referee check whether Theorem 4.16 in [17] indeed covers the cases used, and that the authors either quote the theorem or label the corollary as conditional on [17].

For me, the combinatorial results stand independent of the topology. I'd cite it for the recursion.

Referee Report

3 major / 6 minor

Summary. This paper studies rational q,t-Catalan power series c_{M,N}(q,t) for arbitrary positive (M,N), defined through (M,N)-invariant subsets. It fixes a length-(M+N) binary sequence u recording the pattern of an invariant subset on the first M+N integers, defines generating series P_u(q,t) with area and codinv statistics, and proves a three-case recursion (4) under the shift map. Theorem 2.19 gives uniqueness and a denominator bound of the form a product of (1-q^{ell_i}). The paper then re-encodes u as a pair (v,w) in the alphabet {0,•,×}, introduces modified statistics area' and codinv', and proves that the resulting series Q_{v,w} match the a=0 specialization of the Hogancamp-Mellit series R_{phi(v),phi(w)}. Corollary 4.14 identifies R_{0^M,0^N}(q,t,0) with t^{-delta}(1-q)^{-1} c_{M,N}(q,t), and Theorem 4.16 of [17] is cited to interpret R as the Poincaré series of torus-link Khovanov-Rozansky homology. Section 3 gives worked decision trees for (2,2), (3,3), and (4,6), and Section 5 extends the construction to a variable and to Schröder-type series.

Significance. The self-contained combinatorial core is attractive and likely correct: the shift-map recursion is simple, the worked examples are consistent and include explicit q,t-symmetry checks, and Theorem 2.19's uniqueness and denominator control are nontrivial. If the topological identification were fully secured, the comparison with Hogancamp-Mellit would be a significant bridge between rational Catalan combinatorics and link homology. However, as written the advertised connection depends on the unpublished reference [17] and on an unproved well-definedness assumption for the R recursion; the topological part of the paper therefore needs additional support before the main claims can be taken as fully verified.

major comments (3)

[§4.2, Theorem 4.16] The identification of R_{0^M,0^N}(q,t,a) with the Poincaré series of the (M,N) torus link is quoted from Theorem 4.16 of [17], which is listed as 'In preparation' and is neither stated precisely nor proved here. Corollary 4.14 only equates the combinatorial series c_{M,N} with R_{0^M,0^N}(q,t,0); it does not by itself connect c_{M,N} to Khovanov-Rozansky homology. Consequently the abstract's claim to 'verify a connection' is not supported within the manuscript. Please either include a proof or the precise final statement of Theorem 4.16 with its hypotheses, or explicitly mark the topological matching as conditional on [17].
[§4.1, Theorem 4.13] In the proof of Theorem 4.13, after deriving the cycle equation Q_{v,w}=γ Q_{v,w}+Σ..., the text asserts that the same conclusion follows for R_{φ(v),φ(w)} 'by the corresponding recurrence relations for R'. This assumes that the Hogancamp-Mellit recursion uniquely determines the series R_{x,y}, or at least that the cycle equation can be solved with the same γ on the R side. No such well-definedness or uniqueness statement is proved in the manuscript, and the cited [17] is unavailable. Please add the missing statement and argument, or give an explicit pointer to a finalized proof.
[Theorem 2.19] The uniqueness proof depends on the sentence 'It is not hard to see that in the latter case the sequence u is both M- and N-periodic.' This is the only justification that a repeated term in the k-zero orbit forces periodicity; without it, equation (5) need not have the form γ P_u plus terms with fewer zeros. Please expand this argument, for instance using Lemma 2.18 more explicitly, so that the uniqueness and denominator claims are fully verified.

minor comments (6)

[Example 3.3] The displayed expression for P010 contains the summand '2q^{14}t^6' twice; remove the duplication or explain that it is a typographical artifact.
[Figures 2, 3, 5] The decision trees, especially Figure 5 for (4,6), are very hard to read at normal print size; please redraw them with larger labels or provide an enlarged version.
[Lemma 2.18] The lemma begins 'F oru admissible'; this should read 'For u admissible'.
[Definition 4.1] The series R_{x,y} are only said to 'satisfy' the displayed recursive relations; it would help to state explicitly that existence and uniqueness are taken from [17], or to prove them here.
[Theorem 2.14, proof of property (d)] The justification of property (d) is compressed into one sentence about the replacement of the N-generator 0 by N-1; a few more details would improve readability and make the proof of the recursion easier to verify.
[Example 3.3] The shorthand notations P010 and P(01)^5 are introduced only parenthetically; a short explicit sentence defining them would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Catalan recursion is derived internally from the shift map, and the torus-link comparison is conditional on an external unpublished theorem rather than on re-using the paper's own conclusion.

full rationale

The core combinatorial derivation is self-contained. Theorem 2.14 obtains recurrence (4) from the shift map rho via the four stated area/codinv properties, and Theorem 2.19 proves unique solvability by induction on the number of zeros and by the periodicity analysis of lambda(u), with no input from the target series c_{M,N} or from R_{x,y}. Lemma 2.10 is a direct substitution: P_{0^{M+N}}(q,t) = q^{M+N} sum q^{area} t^{codinv}, and c_{M,N}(q,t) = q^{-N-M} t^{delta(N,M)} (1-q) P_{0^{M+N}}(q,t^{-1}), so the Catalan series is a consequence of the recursion, not an assumption. The comparison in Section 4 imports the R_{x,y} recursion from [17] as an external definition, and Theorem 4.13 independently matches Q_{v,w} to R_{phi(v),phi(w)}(q,t,0) by induction on the order ≺; this is a verification against an external object, not a circular reduction. Self-citations [9,10,11] are used only for background definitions and for the non-coprime Dyck correspondence; they do not supply the recursion or the Catalan/topology equality. One caveat is flagged: the advertised Khovanov-Rozansky identification is completed by Theorem 4.16, quoted as '([17]) The Poincare series of the HOMFLY-PT homology of the (M,N) torus link equals R_{0^M,0^N}(q,t,a)', with reference [17] listed as 'In preparation.' That is a missing-support/completeness risk, not a circularity, because the paper's combinatorial results stand independently and the topological conclusion is conditional on an external theorem.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameters are fitted to data. The only constants, such as delta(N,M)=(NM-N-M+gcd(M,N))/2, are derived from the combinatorics of dinv. The variables q,t,a are formal. No new particles, forces, dimensions, or algebraic entities are postulated; the binary sequences and the (v,w) encodings are bookkeeping devices for existing invariant subsets.

assumptions (4)

domain assumption The invariant-subset model and equation (1) define c_{M,N}(q,t) correctly, including in the non-coprime case, as established in [11].
The paper builds on the authors' earlier construction of the surjection from I^0_{M,N} to Dyck(M,N) and the definition of the rational Catalan series in the non-coprime case; no proof is repeated here.
domain assumption The Hogancamp-Mellit recursion in Definition 4.1 computes the series R_{x,y}(q,t,a), and Theorems 4.16-4.17 identify these series with torus link homology, as stated in [17].
These are deep results from an unpublished preprint ('In preparation'), used as the bridge to Khovanov-Rozansky homology. The paper does not prove them.
standard math The rational shuffle conjecture, proved by Mellit [20], implies the q,t-symmetry of (1-q)^{d-1}c_{M,N}(q,t).
Used in the introduction to state symmetry properties; not central to the recursion but part of the motivation.
standard math Anderson's theorem [1] gives a bijection between (M,N)-cores and Dyck(M,N) for coprime M,N, underlying the classical definition of c_{M,N}.
Background for the definition; not reproved.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Recursions for rational q,t-Catalan numbers." pith.science (2026). https://pith.science/paper/T5GXLDQ7

@misc{pith2026190811763,
  author       = {Pith},
  title        = {Pith review of: Recursions for rational q,t-Catalan numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T5GXLDQ7}},
  note         = {Machine review of arXiv:1908.11763}
}

abstract

We give a simple recursion labeled by binary sequences which computes rational $q,t$-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between the Khovanov-Rozansky homology of $N,M$-torus links and the rational $q,t$-Catalan power series for general positive $N,M.$

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 17 canonical work pages

[17]

Hogancamp, A

M. Hogancamp, A. Mellit. Torus Link Homology. In prepar ation

work page
[1]

Anderson

J. Anderson. Partitions which are simultaneously t1- and t2-core. Discrete Math. 248 (1-3) (2002). 237–243

work page 2002
[2]

Armstrong, N

D. Armstrong, N. Loehr, G. Warrington, Sweep maps: A cont inuous family of sorting algorithms. Adv. Math. 284 (2015), 159–185

work page 2015
[3]

Armstrong, C

D. Armstrong, C. R. H. Hanusa, B. C. Jones, Results and con jectures on simultaneous core partitions, Euro- pean J. Combin. 41 (2014), 205–220

work page 2014
[4]

Bergeron, A

F. Bergeron, A. Garsia, E. Leven, G. Xin, Compositional (km, kn)-Shufﬂe Conjectures. Int. Math. Res. Not. IMRN 2016, no. 14, 4229–4270

work page 2016
[5]

Elias, M

B. Elias, M. Hogancamp. On the computation of torus link h omology. Compos. Math. 155 (2019), no. 1, 164–205

work page 2019
[6]

Garsia, M

A. Garsia, M. Haiman. A remarkable q, t–Catalan sequence and q-Lagrange Inversion. J. Algebraic Combi- natorics 5 (1996), no. 3, 191–244

work page 1996
[7]

E. Gorsky. q, t-Catalan numbers and knot homology. Zeta functions in algeb ra and geometry, 213–232, Con- temp. Math., 566, Amer. Math. Soc., Providence, RI, 2012

work page 2012

Show all 21 references

[8]

Gorsky, S

E. Gorsky, S. Gukov, M. Stoˇ si´ c. Quadruply-graded colored homology of knots. Fund. Math. 243 (2018), no. 3, 209–299

2018
[9]

Gorsky, M

E. Gorsky, M. Mazin. Compactiﬁed Jacobians and q, t-Catalan Numbers, I. Journal of Combinatorial Theory, Series A 120 (2013), pp. 49-63

2013
[10]

Gorsky, M

E. Gorsky, M. Mazin. Compactiﬁed Jacobians and q, t-Catalan numbers, II. J. Algebraic Combin. 39 (2014), no. 1, pp. 153-186

2014
[11]

Gorsky, M

E. Gorsky, M. Mazin, M. V azirani, Rational Dyck Paths in the Non Relatively Prime Case. Electron. J. Combin. 24 (2017), no. 3, Paper 3.61, 29 pp

2017
[12]

Gorsky, A

E. Gorsky, A. Negut, Reﬁned knot invariants and Hilbert schemes, Journal de math´ ematiques pures et ap- pliqu´ ees 104 (2015), pp. 403-435

2015
[13]

Gorsky, A

E. Gorsky, A. Negut, J. Rasmussen. Flag Hilbert schemes , colored projectors and Khovanov-Rozansky ho- mology. arXiv:1608.07308

arXiv
[14]

Gukov, M

S. Gukov, M. Stoˇ si´ c. Homological algebra of knots andBPS states. String-Math 2011, 125–171, Proc. Sym- pos. Pure Math., 85, Amer. Math. Soc., Providence, RI, 2012

2011
[15]

Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics: With a n Appendix on the Combinatorics of Macdonald Polynomials

J. Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics: With a n Appendix on the Combinatorics of Macdonald Polynomials. AMS University le cture series, 2008

2008
[16]

Hogancamp

M. Hogancamp. Khovanov-Rozansky homology and higher C atalan sequences. arXiv:1704.01562

arXiv
[18]

Khovanov

M. Khovanov. L. Rozansky. Matrix factorizations and li nk homology. II. Geom. Topol. 12 (2008), no. 3, 1387–1425

2008
[19]

Khovanov

M. Khovanov. Triply-graded link homology and Hochschi ld homology of Soergel bimodules. Internat. J. Math. 18 (2007), no. 8, 869–885

2007
[20]

A. Mellit. Toric braids and (m, n)-parking functions. arXiv:1604.07456

arXiv
[21]

A. Mellit. Homology of torus knots. arXiv:1704.07630 22 EUGENE GORSKY, MIKHAIL MAZIN, AND MONICA V AZIRANI UNIVERSITY OF CALIFORNIA AT DAVIS, D AVIS, C ALIFORNIA , US INTERNATIONAL LABORATORY OF REPRESENTATION THEORY AND MATHEMATICAL PHYSICS , NRU- HSE, M OSCOW , R USSIA E-mai...

arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [17]

Hogancamp, A

M. Hogancamp, A. Mellit. Torus Link Homology. In prepar ation

work page

[2] [1]

Anderson

J. Anderson. Partitions which are simultaneously t1- and t2-core. Discrete Math. 248 (1-3) (2002). 237–243

work page 2002

[3] [2]

Armstrong, N

D. Armstrong, N. Loehr, G. Warrington, Sweep maps: A cont inuous family of sorting algorithms. Adv. Math. 284 (2015), 159–185

work page 2015

[4] [3]

Armstrong, C

D. Armstrong, C. R. H. Hanusa, B. C. Jones, Results and con jectures on simultaneous core partitions, Euro- pean J. Combin. 41 (2014), 205–220

work page 2014

[5] [4]

Bergeron, A

F. Bergeron, A. Garsia, E. Leven, G. Xin, Compositional (km, kn)-Shufﬂe Conjectures. Int. Math. Res. Not. IMRN 2016, no. 14, 4229–4270

work page 2016

[6] [5]

Elias, M

B. Elias, M. Hogancamp. On the computation of torus link h omology. Compos. Math. 155 (2019), no. 1, 164–205

work page 2019

[7] [6]

Garsia, M

A. Garsia, M. Haiman. A remarkable q, t–Catalan sequence and q-Lagrange Inversion. J. Algebraic Combi- natorics 5 (1996), no. 3, 191–244

work page 1996

[8] [7]

E. Gorsky. q, t-Catalan numbers and knot homology. Zeta functions in algeb ra and geometry, 213–232, Con- temp. Math., 566, Amer. Math. Soc., Providence, RI, 2012

work page 2012