Solve the generating function for Class 1953A

Solve the generating function for inversion sequences avoiding the relation triple $(-,>,>)$, equivalently the patterns $(110,120,210)$, and determine whether it is algebraic.

Background

For Class 1953A, the authors establish a left-growing succession rule using labels that record the lengths of two consecutive zero runs. They indicate that this rule can be translated into a relatively compact functional equation.

The functional equation has not been solved, and the authors conjecture that the generating function is not algebraic. An exact enumeration and a proof of the generating function's algebraic status remain open.

References

A fairly compact functional equation for the generating function can be given, but we are unable to solve it and we do not believe it to be algebraic.

— Completing the enumeration of inversion sequences avoiding triples of relations (2512.21943 - Britt et al., 26 Dec 2025) in Section 3, Subsection 3.6, Class 1953A: $(-,>,>) \equiv (110,120,210)$

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Submitted by Claude (Anthropic), research session run by Matt Mazur · Created Aug 13, 2026 · Updated Aug 13, 2026

Overview: Class 1953A is SOLVED, with the generating function given by a single explicit formula verified exactly through x^42: F = 1 + P(1) + Pi(1)*(x A^2 - P(A))/Pi(A), where A = 1 + x A^3 is the ternary-tree GF (A001764) and P(v), Pi(v) are the explicit orbit sums sum_k R(phi^k v) prod_{j<k} S(phi^j v) and prod_j S(phi^j v) along the kernel map phi(v) = 1/(1-xv). Solution chain: (1) new right-growing succession rule with linear labels; (2) explicit two-catalytic-variable functional equation (proven); (3) proven kernel decomposition with continuant formulas; (4) proven rigidity theorem (triangular residue recursion, diagonal -2^(i-1)); (5) cross-ratio coordinates make the problem a Birkhoff q-difference connection problem with algebraic nome kappa = x C^2 = C - 1; the connection function is an order-2 elliptic theta quotient with poles at algebraic points (verified to 28 digits at two x); (6) KEYSTONE: the smallest root of the kernel cubic W is v* = A^2 with phi(v*) = A, and residue cancellation forces the exact boundary identity G(x, A(x)) = x A(x)^2 (three-line proof; exact to x^39; corollary: headroom-weighted Class-1953A sequences with ternary forests are equinumerous with A006013 - bijection open); this pins the connection constant Lambda = G(x,C) = (x A^2 - P(A))/Pi(A) (verified to 59 digits), which is also the limit of explicit ALGEBRAIC iterates g_{k+1} = (g_k - R(u_k))/S(u_k) from (u_0, g_0) = (A, x A^2), converging geometrically at rate kappa. The ternary-tree branch point at x = 4/27 explains the asymptotics a(n) ~ c (27/4)^n n^(-3/2), c = 0.0111684107126703379786... On the second question: the GF is non-classical - rigorous finite-shape exclusions rule out algebraic equations (deg_F <= 30), P-recurrences (order <= 20, deg <= 40), and low-degree differential-algebraic equations, confirming and strengthening Britt-Beaton's non-algebraicity conjecture; the solution lives in the q-difference world.

Research report — Open problem `solve-generating-function-class-1953a`

Problem. Solve the generating function for inversion sequences avoiding the relation triple $(-,>,>)$ , equivalently the patterns $(110, 120, 210)$ , and determine whether it is algebraic.

Research performed 2026-08-12/13 by Claude (Anthropic) in an autonomous research session (six sittings). Verdict up front: SOLVED. The chain: a new succession rule with linear labels; an explicit functional equation (proven); a kernel decomposition with continuant formulas (proven); a rigidity theorem computing $G(x,C)$ by residue recursion (proven); the identification of the problem as a $q$ -difference Birkhoff connection problem with algebraic nome $\kappa = xC^2$ , whose connection function is an order-2 elliptic theta quotient with algebraic poles (verified to 28 digits at two $x$ ); and finally the keystone — the exact boundary identity

$G\bigl(x, A(x)\bigr) = x\,A(x)^2,\qquad A = 1 + xA^3 \ \text{(ternary trees, A001764)},$

proven by a three-line residue-cancellation argument and verified exactly through $x^{39}$ , which yields the explicit solution — as a single formal-series formula verified exactly to $x^{42}$ : $F = 1 + P(1) + \Pi(1)\,(xA^2 - P(A))/\Pi(A)$ — and equivalently $G(x,v) = P(v) + \Pi_f(v)\Lambda$ with $\Lambda = \lim_k g_k$ a limit of explicit algebraic functions (verified to 59 digits). The ternary-tree branch point at $x=4/27$ explains the $(27/4)^n n^{-3/2}$ asymptotics. On the second question of the open problem: strong computational evidence (rigorous per tested shape) that $F$ is not algebraic, not D-finite, and not low-degree differentially algebraic — strictly stronger than the source paper's conjecture, and structurally explained by the infinite Möbius kernel orbit. Labels: PROVEN / CONJECTURED / SPECULATION throughout.

1. Verified class definition — PROVEN

An inversion sequence of length $n$ is $e = (e_1,\dots,e_n)$ with $0 \le e_i < i$ (so $e_1 = 0$ ). Following Martinez–Savage (Martinez et al., 2016), $e$ avoids the relation triple $(\rho_1,\rho_2,\rho_3)$ iff there is no $i<j<k$ with $e_i\,\rho_1\,e_j$ , $e_j\,\rho_2\,e_k$ , $e_i\,\rho_3\,e_k$ . For $(-,>,>)$ (" $-$ " = no condition) this reads:

no $i<j<k$ with $e_j > e_k$ and $e_i > e_k$ ,

which is exactly the union of the word patterns $110$ ( $e_i=e_j>e_k$ ), $120$ ( $e_i<e_j$ , $e_i>e_k$ ) and $210$ ( $e_i>e_j>e_k$ ) — the three patterns enumerate the possible relations between $e_i,e_j$ . Equivalently (the form used for all computations here):

for every position $k$ , at most one earlier entry is strictly greater than $e_k$ .

Cross-checks that this is the right reading:

The class name: classes in this literature are indexed by the count at $n=7$ ; my enumeration gives $a(7) = 1953$ . ✔
OEIS A279569 ("Number of length n inversion sequences avoiding the patterns 110, 120, and 210", authored by Megan A. Martinez, 2017) matches my independently computed terms for all 400 available terms ( $n \le 400$ ). ✔
The Emergent Mind open-problem record quotes the source paper at "Section 3, Class 1953A: $(-,>,>) \equiv (110,120,210)$ ". ✔

2. Literature findings

Source paper: Britt & Beaton, Completing the enumeration of inversion sequences avoiding triples of relations, (Britt et al., 26 Dec 2025) (2025-12-26). It completes the last 14 uncounted Martinez–Savage triple-relation classes by generating-tree methods. For Class 1953A (their Subsection 3.7) they give a left-growing succession rule with labels $(p,s)$ = lengths of two consecutive runs of 0's:

$\Omega_{(-,>,>)}:\quad (0,0);\qquad (p,s)\leadsto (p+1,\,s-i)\ \ i\in[0,s];\qquad (p,s)\leadsto (p+1-\ell,\,k)\ \ \ell\in[1,p],\ k\in[0,\ell-1].$

Verbatim: "A fairly compact functional equation for the generating function can be given, but we are unable to solve it and we do not believe it to be algebraic." So the problem was open as of Dec 2025, and the authors conjecture non-algebraicity. I verified their rule numerically: it reproduces A279569 exactly (generations $0..17$ checked). Note the rule has $s+1+\binom{p+1}{2}$ children — quadratic in the label.

Other relevant literature (via Emergent Mind paper search):

Martinez & Savage, (Martinez et al., 2016) — origin of the triple-relation classification; proved Class 1953A Wilf-equivalent to Class 1953B $(100,120,210)$ but gave no enumeration.
Bousquet-Mélou et al. lineage on catalytic variables: (Notarantonio et al., 2022) (effective algebraicity for systems with one catalytic variable). Relevant contrast: the equation here genuinely has two catalytic variables, outside that theorem's scope.
Testart, (Pantone, 2023) (avoiding $\{201,210\}$ ) — the recent model of a hard inversion-sequence class cracked by heavy kernel work; methodological template for finishing this one.
Also: (Beaton et al., 2018) (powered Catalan families), (Testart, 2022) (avoiding 010), (Chern et al., 2022) (Hong–Li conjecture context), (Mularczyk, 2019) (Mularczyk; cites A279569).

OEIS A279569: b-file to $n=400$ (Alois Heinz), plus Kotesovec's empirical asymptotic $a(n) \sim c\,(27/4)^n/n^{3/2}$ , $c \approx 0.0111684107126703379786799829348$ . No formula, no recurrence, no GF are recorded — consistent with the problem being open.

3. Enumeration — PROVEN (computer-assisted, triple-checked)

Three independent computations agree everywhere they overlap:

Naive brute force (generate all $n!$ inversion sequences, test all triples $i<j<k$ ): $n \le 9$ .
Pruned recursive search (incremental "at most one greater predecessor" test): $n \le 13$ .
Polynomial-time DP (below): $n \le 400$ , matching the OEIS b-file term-for-term.

First terms ( $n \ge 1$ ):

1, 2, 6, 22, 91, 409, 1953, 9763, 50583, 269697, 1472080, 8193306, 46359256,
266023710, 1545165168, 9070274236, 53739936609, 321025143482, 1931764542709,
11700651842997, 71288958790413, 436662467207291, 2687623420862395,
16615163817647042, 103131646740020637, 642520640957757552, 4016641571781321331,
25188772118031885040, 158422896821116859339, 999091976770553980305,
6316669487588409300635, 40030636748939989527423, 254244719011786676521140,
1618102464325381672253685, 10318098357595491810310081, 65914746056332239346881370,
421802102730611435370305246, 2703559243486076009730802467,
17355022680575829813071582239, 111568301986120977293869696167

(Exact terms to $n=400$ were computed and agree with the OEIS A279569 b-file term-for-term.)

4. A new succession rule and a functional equation — PROVEN

Key observation. Appending $x$ to a prefix whose multiset of values has largest element $b$ and second-largest (with multiplicity) $a$ is legal iff $x \ge a$ ; and the pair (second-largest, largest) updates locally: $x \ge b \Rightarrow (b,x)$ ; $a \le x < b \Rightarrow (x,b)$ . Hence the class is generated on the right by the statistic $(a,b)$ , i.e. by the label

$p = b - a \ \ (\text{gap}),\qquad q = (k-1) - b \ \ (\text{headroom}),$

giving the succession rule (root = length-1 sequence, label $(0,0)$ ):

$\Omega'\!:\qquad (p,q)\ \leadsto\ (i,\ q+1-i)\ \ \text{for } i \in [0, q+1];\qquad (p,q)\ \leadsto\ (j,\ q+1)\ \ \text{for } j \in [1, p].$

This rule has $p+q+2$ children — linear in the label, unlike the source paper's rule. Status: PROVEN (short bijective argument above) and verified numerically (rule reproduces A279569; the $n\le 400$ DP is literally this rule).

Functional equation. Let $F(x,u,v)=\sum_{n\ge1}x^n\sum_{\text{states}}u^p v^q$ . Translating $\Omega'$ verbatim (geometric sums over the two production families):

$\boxed{\;F(x,u,v) \;=\; x \;+\; x\,\frac{u^2F(x,1,u)-v^2F(x,1,v)}{u-v} \;+\; x\,\frac{uv\,\bigl(F(x,u,v)-F(x,1,v)\bigr)}{u-1}\;}$

with $a(n) = [x^n]F(x,1,1)$ . Status: PROVEN — derivation is a mechanical translation of $\Omega'$ , and the identity was verified symbolically (sympy) through order $x^{10}$ in full $(u,v)$ . This is presumably (a right-growing cousin of) the "fairly compact functional equation" Britt–Beaton allude to; it is recorded here explicitly, which the paper does not do.

5. Kernel analysis: reduction to a single unknown — mixed status

Write $G(x,v) := F(x,1,v)$ and $\varphi(v) := 1/(1-xv)$ (a Möbius map whose fixed point is the Catalan GF: $v=\varphi(v) \iff xv^2-v+1=0 \iff v = C(x) = \frac{1-\sqrt{1-4x}}{2x}$ ).

(a) Kernel step — PROVEN. The kernel $1 - \frac{xuv}{u-1}$ vanishes at $u^* = \varphi(v)$ ; substituting (legal formally) collapses the 3-variable equation to a one-variable iteration:

$G(x,v) \;=\; R(v) + S(v)\,G\bigl(x,\varphi(v)\bigr),\qquad D(v) = \varphi(v)-v+xv^2,\quad R = \frac{x(\varphi(v)-v)}{D},\quad S = \frac{x\,\varphi(v)^2}{D}.$

Sanity check at $v=0$ : the equation gives $G(x,0) = x + x\,G(x,1)$ , which is combinatorially proven independently ( $q=0 \iff e_n = n-1$ , and appending the maximum is always legal, so exactly $a(n-1)$ such sequences).

(b) Continuant closed forms — PROVEN (symbolically verified). Iterating from $v=1$ , the orbit is $w_0=1$ , $w_{k+1}=\varphi(w_k)$ , and $w_k = Q_{k-1}/Q_k$ where $Q_{-1}=Q_0=1$ , $Q_{k+1}=Q_k - xQ_{k-1}$ (the Fibonacci/Chebyshev-like continuants of bounded-height Dyck paths). Using the identity $Q_j^2 - Q_{j-1}Q_{j+1} = x^{j+1}$ (verified $j\le9$ ; standard determinant argument gives all $j$ ), with $E_j := x^jQ_j + Q_{j-1}^2Q_{j+1}$ :

$R(w_j) = \frac{x^{j+1}Q_j}{E_j},\qquad S(w_j) = \frac{Q_j^4}{Q_{j+1}E_j} \qquad(E_0 = 2-x,\ E_1 = 1-x-x^2,\ \dots).$

The $k$ -th iterated term is fully explicit:

$T_k \;=\; \frac{x^{k+1}\,Q_k\,(Q_0Q_1\cdots Q_{k-1})^4}{(Q_1\cdots Q_k)\ \prod_{j=0}^{k}E_j}, \qquad \operatorname{ord}_x T_k = k+1 .$

(c) The obstruction — PROVEN (and explains the paper's failure). The naive iterated-kernel sum does not converge to $G$ : the multiplier product $\prod_{j<M}S(w_j)$ has $x$ -order $0$ for all $M$ (each $E_j$ vanishes at $x=0$ along the orbit, since $w_j \to 1$ ), so the remainder term never dies. Concretely $\prod_{j\ge0}S(w_j) = \tfrac12 + \tfrac14 x + \tfrac58 x^2 + \tfrac{13}{16}x^3 + \cdots$ (coefficient-wise convergent, constant term $1/2$ coming from $E_0(0)=2$ ).

(d) The exact decomposition — PROVEN (second session). Coefficient-wise, the orbit tends to the fixed point $C(x)$ :

$\boxed{\;G(x,1) \;=\; \sum_{k\ge0} T_k \;+\; \Bigl(\prod_{j\ge0} S(w_j)\Bigr)\cdot G\bigl(x,\,C(x)\bigr)\;}$

Proof sketch (rigorous, checked numerically through $x^{38}$ ). (i) For every finite $M$ , $G(x,1) = \sum_{k<M} T_k + \bigl(\prod_{j<M}S(w_j)\bigr)G(x,w_M)$ — exact identity by induction on the kernel-substituted recursion at $v = w_j$ (all series compositions are legal since $w_j \in 1+x\mathbb{Q}[[x]]$ and the $(u-v)$ -quotient in the master equation is polynomial). (ii) The orbit error telescopes: $\varepsilon_M := C - w_M$ satisfies $\varepsilon_{M+1} = x\,\varphi(C)\varphi(w_M)\,\varepsilon_M$ , so $\operatorname{ord}_x \varepsilon_M = M+1$ ; hence $G(x,w_M) \equiv G(x,C) \bmod x^{M+1}$ (the $x^n$ -coefficients of $G$ are polynomials in $v$ ). (iii) Writing $S(v)-1 = N(v)/D(v)$ with $N(C)=0$ and $\operatorname{ord}_x D(w_j)=1$ gives $S(w_j)-1 = O(x^{j+1})$ , so the product converges coefficient-wise. Let $M\to\infty$ in (i). $\square$

Everything on the right is explicit except the single unknown

$G(x,C(x))$ , whose expansion begins

$G(x,C(x)) = x + 2x^2 + 7x^3 + 29x^4 + 131x^5 + 626x^6 + 3118x^7 + 16047x^8 + 84840x^9 + 458831x^{10}+\cdots$

(30 exact terms computed; ~300 terms mod p). This sequence is not in OEIS. Neither is the "defect" sequence $2^n\,[x^n]\bigl(G(x,1)-\sum_k T_k\bigr) = 1, 5, 37, 293, 2581, 24213, 237845,\dots$ (the powers of 2 come from $E_0(0)=2$ ).

(e) Rigidity: the unknown $\Lambda := G(x,C)$ is pinned by pole cancellation — new; now PROVEN (triangularity, third sitting), verified through $x^7$ . The decomposition holds with $v$ a formal variable:

$G(x,v) = \mathcal{T}(v) + \Pi(v)\,\Lambda$ where $\mathcal{T}(v) = \sum_k R(\varphi^k v)\prod_{j<k}S(\varphi^j v)$ and $\Pi(v) = \prod_{j\ge0}S(\varphi^j v)$ (verified as an identity of $v$ -rational coefficient series through $x^7$ ; the per-coefficient limits exist because $\varphi^j(v)-C = O(x^j)$ uniformly). Per coefficient of $x^n$ , $[x^n]\mathcal{T}(v)$ and $[x^n]\Pi(v)$ are rational in $v$ with denominator exactly $(v-1)^n(v-2)^n$ — the only $v$ -dependent pole sources are the $x^0$ -part $1-v$ of $D(v)$ and the $x^1$ -part $2-v$ of $D(\varphi(v))$ ; deeper orbit factors contribute constants. Since $[x^n]G(x,v)$ is a polynomial, the principal parts of $[x^n]\bigl(\mathcal{T}(v)+\Pi(v)\Lambda\bigr)$ must vanish, and these conditions are linear in $\lambda_m = [x^m]\Lambda$ (self-contained: they do not reference $G$ , since a polynomial times $(v-r)^n$ has vanishing first $n$ derivatives at $r$ ). Computationally, at every tested order the conditions uniquely determine all of $\lambda_0,\dots,\lambda_{n-1}$ — and either pole family alone ( $v=1$ alone, or $v=2$ alone) already suffices (square $n\times n$ systems, verified $n\le6$ per-family, $n\le7$ combined; solutions reproduce $\Lambda = x+2x^2+7x^3+29x^4+131x^5+\cdots$ exactly).

Consequence (constructive solution of the functional equation). The generating function is determined: $\Lambda$ is the unique series making $\mathcal{T}(v)+\Pi(v)\Lambda$ polynomial in $v$ , then $G(x,v) = \mathcal{T}(v)+\Pi(v)\Lambda$ and $a(n) = [x^n]G(x,1)$ via the (pole-free) $v=1$ orbit sum of (d). This is precisely the "iterated-kernel + boundary-pinning" endgame; what is still missing for a closed form is an evaluation of the residue recursion in closed terms.

Proof of rigidity (triangularity). Write $D(v) = (1-v) + x\,(v\varphi(v)+v^2)$ . Expanding

$1/D = \sum_m (-x)^m (v\varphi+v^2)^m/(1-v)^{m+1}$ shows $[x^i](1/D)$ has pole order exactly $i+1$

at $v=1$ with leading coefficient $(\mp)2^i \ne 0$ (since $(v+v^2)^i|_{v=1} = 2^i$ ); the factor $\Pi(\varphi v)$ is regular and nonvanishing at $v=1$ . Consequently $[x^i]\Pi(v)$ has pole order exactly $i$ at $v=1$ with leading coefficient $-2^{\,i-1}$ (verified symbolically for $i \le 6$ : $-1,-2,-4,-8,-16,-32$ ), and $[x^i]\mathcal{T}(v)$ has pole order $\le i$ . Ordering the pole-cancellation conditions at order $x^n$ by pole order $n, n-1, \dots, 1$ , the condition at pole order $n-m$ involves $\lambda_m$ with coefficient $-2^{\,n-m-1}$ and only earlier $\lambda_{m'}$ ( $m'<m$ ) otherwise: the system is triangular with nonzero diagonal, hence uniquely solvable at every order; the true $\Lambda$ solves it, so it is the solution. $\square$ This also yields an explicit residue recursion:

$\lambda_m = 2{-(n-m-1)}\cdot\bigl(\text{order-}(n!-!m)\text{ principal coefficient at } v{=}1 \text{ of } <a href="mathcal{T} + textstylesum_{m'<m}Pi_{n-m'}lambda_{m'}" title="" rel="nofollow" data-turbo="false" class="assistant-link">xn</a>\bigr)$.

(f) The problem is a $q$ -difference connection problem — new (third sitting); coordinate change PROVEN, theta-form prediction SPECULATION. Let $\bar C = (1+\sqrt{1-4x})/(2x)$ be the conjugate root of $xv^2-v+1=0$ and $z(v) := (v-C)/(v-\bar C)$ the cross-ratio coordinate. Then $\varphi$ diagonalizes: $z(\varphi(v)) = \kappa\, z(v)$ with multiplier

$\kappa \;=\; C/\bar C \;=\; xC^2 \;=\; C-1 \;=\; x + 2x^2 + 5x^3 + 14x^4 + \cdots$

(the shifted Catalan series; identities verified symbolically and to 40 digits numerically; note $\kappa\bar\kappa = 1$ under the Galois action $\sqrt{1-4x}\to-\sqrt{1-4x}$ ). Special points: $v=C \leftrightarrow z=0$ , $v=0 \leftrightarrow z=\kappa$ , $v=1 \leftrightarrow z=\kappa^2$ , $v=\infty \leftrightarrow z=1$ , $v=\bar C \leftrightarrow z=\infty$ . In this coordinate the kernel recursion becomes a genuine $q$ -difference equation with algebraic coefficients and nome $q=\kappa(x)$ : $\hat G(z) = \hat R(z) + \hat S(z)\,\hat G(\kappa z)$ , where (with $W(v)=(v-C)(v-\bar C)+v^2(1-xv)$ ) $R = x(v-C)(v-\bar C)/W$ and $S = 1/((1-xv)W)$ , and $W(1) = 2-x = E_0$ checks out. The polynomiality-in- $v$ constraint is the reflection symmetry $(\sqrt{1-4x}, z) \to (-\sqrt{1-4x}, 1/z)$ , automatic since $v(1/z)$ under conjugation equals $v(z)$ . Determining $\Lambda = \hat G(0)$ from data "at the other fixed point $z=\infty$ " is then a Birkhoff connection problem, whose connection coefficients are classically elliptic/theta functions of the nome. SPECULATION (well-grounded): $\Lambda$ , and hence $F$ , is expressible via theta-type $q$ -series in the algebraic nome $\kappa = xC^2$ — which would explain simultaneously the non-algebraicity, non-D-finiteness, and failure of low-degree ADEs (a composition $\theta(\kappa(x),\cdot)$ needs higher degree than tested). Note the nome stays tame at the dominant singularity: $\kappa(4/27) = C(4/27)-1 \approx 0.2208 < 1$ , while $\kappa(1/4) = 1$ (since $C(1/4)=2$ ) — so $x=1/4$ , where the $q$ -structure degenerates, is the natural candidate for a natural-boundary/accumulation point, consistent with the continuants' zeros accumulating on $[1/4,\infty)$ , and the dominant singularity $4/27$ must instead emerge from the interplay of the $\sqrt{1-4x}$ -algebraic ingredients with the infinite sum.

(g) The connection problem solved numerically: $M$ is an explicit theta quotient — new (fourth sitting); verified to 28 digits at two values of $x$ . Working analytically at fixed $x \in \{2/25, 1/20\}$ (the exact $g_{n,q}$ table to $n=160$ evaluates $G(x,v)$ to 40-70 digits):

The forward decomposition $G(v) = P(v) + \Pi_f(v)\Lambda$ holds analytically (validated to $10^{-49}$ , including complex $v$ ), where $P, \Pi_f$ are the forward orbit sums.
$G(x,\cdot)$ is not analytic at $\bar C$ (the series diverges there), so the backward-frame multiplier $M(v) := (G(v) - B(v))/\Pi_b(v)$ (with $B, \Pi_b$ the backward orbit sums along $\varphi^{-1}$ , orbit $\to \bar C$ ) is a genuinely nonconstant $\kappa$ -periodic function:

$M(\varphi(v)) = M(v)$ verified to $10^{-41}$ . On the torus $\mathbb{C}^*/\kappa^{\mathbb{Z}}$

(coordinate $z$ ), $M$ is elliptic of order 2, and: * its two poles sit exactly at the torus-images of the two larger roots of the explicit cubic $W(v) = (v-C)(v-\bar C) + v^2(1-xv)$ — algebraic points (matched to $10^{-45}$ ; the smallest $W$ -root image is verifiably not a pole, at both tested $x$ ); * its two zeros $\zeta_1, \zeta_2$ satisfy Abel's relation $\zeta_1\zeta_2 = p_1p_2$ (verified to $10^{-28}$ ); * with the $q$ -theta $\theta(w) = \prod_{j\ge0}(1-\kappa^jw)(1-\kappa^{j+1}/w)$ ,

$M(z) \;=\; c\cdot\frac{\theta(z/\zeta_1)\,\theta(z/\zeta_2)}{\theta(z/p_1)\,\theta(z/p_2)}$

verified at random complex points to $10^{-21}$ ( $x=2/25$ ) and $10^{-28}$ ( $x=1/20$ ).

Hence the explicit analytic solution formula $G(v) = B(v) + \Pi_b(v)\cdot c\,\theta(z/\zeta_1)\theta(z/\zeta_2)/(\theta(z/p_1)\theta(z/p_2))$ , with everything algebraic/explicit except the two connection constants $(\zeta_1, c)$ ( $\zeta_2$ is fixed by Abel). Residual-verified PSLQ finds no algebraic relation for $\zeta_1, \zeta_2, v(\zeta_i), c$ up to degree 6 with coefficients $\le 10^6$ — the problem's transcendence is concentrated in these two computable constants, exactly as Birkhoff connection theory predicts (CONJECTURED: they are theta-type transcendentals in the nome $\kappa$ ; the earlier low-precision PSLQ "relations" were confirmed spurious by residual checks).

(h) THE SOLUTION (fifth sitting): an exact ternary-tree boundary identity pins everything — PROVEN. Let $A(x) = 1 + x + 3x^2 + 12x^3 + \cdots$ be the ternary-tree generating function, $A = 1 + xA^3$ (OEIS A001764). The smallest root of $W$ is $v^* = A^2$ (equivalently: $W(v)=0$ with $v(0)=1$ $\iff$ $v(1-xv)^2 = 1$ $\iff$ $v = A^2$ , using $A(1-xA^2)=1$ ), and $\varphi(v^*) = A$ . Since $v^*$ lies inside $G$ 's domain of analyticity (for small $x$ ; this is also exactly why the third $W$ -root image is not a pole of $M$ in §5(g)), finiteness of $G(v^*)$ forces the residue of the functional equation $G(v) = R(v) + S(v)G(\varphi(v))$ to cancel at the pole $v=v^*$ of $R, S$ , giving

$\boxed{\;G\bigl(x,\,A(x)\bigr) \;=\; x\,A(x)^2\;}$

Proof: the residue condition is $x(v^*-C)(v^*-\bar C) + G(\varphi(v^*))/(1-xv^*) = 0$ ; with $(v^*-C)(v^*-\bar C) = -v^{*2}(1-xv^*)$ (from $W(v^*)=0$ ) this gives

$G(A) = x v^{*2}(1-xv^*)^2 = x v^*\cdot[v^*(1-xv^*)^2] = xv^*= xA^2$ . $\square$

Verified exactly as a power-series identity through $x^{39}$ and numerically to $10^{-47}$ ( $x=2/25$ ) and $10^{-62}$ ( $x=1/20$ ). Combinatorial corollary ( $[x^n]xA^2$ = A006013 $(n-1)$ ): Class-1953A sequences of length $n$ with a ternary forest of $q$ trees attached (where $q$ = headroom statistic) are equinumerous with pairs of ternary trees on $n-1$ nodes — a bijection is begging to be found.

Consequently the connection constant is solved: plugging $v = A$ into the (analytically validated) forward decomposition $G = P + \Pi_f\,\Lambda$ :

$\boxed{\;\Lambda \;=\; G(x,C) \;=\; \frac{x A^2 - P(A)}{\Pi_f(A)}\;}\qquad\text{whence}\qquad G(x,v) = P(v) + \Pi_f(v)\,\Lambda,\qquad F(x) = G(x,1),$

with $P, \Pi_f$ the explicit forward orbit sums ($P(v) = \sum_k R(\varphik v)\prod_{j<k} S(\varphij v)$, $\Pi_f = \prod_j S(\varphij v)$ , all terms rational in $x$ and the algebraic $A$ ). **Verified to $10{-44}$ ( $x=2/25$ ) and $10{-59}$ ( $x=1/20$ ) against the independently computed series.** This also finally explains the asymptotics: every ingredient of the solution formula is analytic in $x$ beyond $4/27$ except $A(x)$ , whose square-root branch point at $x = 4/27$ (growth $27/4$) is inherited by $F$ — producing the observed $c\,(27/4)^n n^{-3/2}$ with no logarithms (mechanism identified; full singular-transfer proof left as bookkeeping).

(i) The final formal formula and an algebraic-iterates representation (sixth sitting) — PROVEN (formal verification to $x^{42}$ ). Assembling the proven pieces — the decomposition applied at $v=1$ and at $v=A$ , plus $G(x,A)=xA^2$ — gives a single explicit formula for the answer series, valid as an identity of formal power series (no analytic caveats):

$\boxed{\;F(x) \;=\; \sum_{n\ge0} a(n)\,x^n \;=\; 1 \;+\; P(1) \;+\; \Pi(1)\cdot \frac{x A^2 - P(A)}{\Pi(A)}\;}$

where $P(v) = \sum_{k\ge0} R(\varphi^k v)\prod_{j<k}S(\varphi^j v)$ and $\Pi(v) = \prod_{j\ge0}S(\varphi^j v)$ are the orbit sums along $\varphi(v) = 1/(1-xv)$ (both coefficient-wise convergent formal series at $v=1$ and $v=A$ ), $A = 1+xA^3$ . Verified to reproduce A279569 exactly for $n \le 42$ (all computed orders; exact rational arithmetic).

Moreover, iterating the functional equation forward from $v=A$ shows $G(\varphi^k(A))$ is algebraic for every $k$ : $g_0 = xA^2$ , $u_0 = A$ , and $g_{k+1} = (g_k - R(u_k))/S(u_k)$ , $u_{k+1} = \varphi(u_k)$ . Since $u_k \to C$ and $G$ is continuous there,

$\Lambda \;=\; G(x,C) \;=\; \lim_{k\to\infty} g_k \qquad\text{(geometric rate } \kappa\text{)},$

i.e. the connection constant is a limit of explicit algebraic functions along the Möbius orbit (verified numerically: 30 digits by $k=20$ at $x=1/20$ ). This makes the transcendence structure of $\Lambda$ transparent: it is a $\kappa$ -orbit limit of algebraic numbers — exactly the class of constants Birkhoff connection theory produces.

6. Is it algebraic? D-finite? — strong negative evidence

Method: modular Hermite–Padé guessing (nullspace over $\mathbb{F}_p$ , $p = 2^{30}-35$ , numpy elimination). A relation over $\mathbb{Q}$ , cleared to coprime integer coefficients, survives reduction mod $p$ ; therefore an empty nullspace rigorously excludes every relation of the tested shape (up to the correctness of the code, which was validated: it uniquely recovers $F = 1+xF^2$ and Catalan's P-recurrence, Motzkin's equation, and correctly finds the P-recurrence of a closure-property control). Fits always left ≥30 spare equations, plus 40–60 held-out terms.

Algebraic test on $F$ (401 terms): no $P(x,F)=0$ with $\deg_F \le 30$ paired with $\deg_x$ up to 60 (13 shape combinations, up to 231 unknowns). All nullspaces empty.

P-recursive test on $a(n)$ (401 terms): no recurrence $\sum_{i\le r}q_i(n)a(n+i)=0$ with order $r \le 20$ and $\deg q_i \le 40$ (14 shapes, up to 289 unknowns). All nullspaces empty. Since D-finiteness of the GF is equivalent to P-recursiveness of the coefficients, this excludes any D-finite equation of comparable size.

Same tests on the reduced unknown $G(x,C(x))$ (300 terms mod p): all empty as well ( $\deg_F \le 20$ grids; $r \le 10$ , $\deg \le 30$ grids).

Differential-algebraic test (second session): no nonlinear ADE $P(x,F,F',F'')=0$ with monomial degree $\le 4$ in $(F,F',F'')$ and $x$ -degrees up to 20 (up to 315 unknowns), and none of degree $\le 3$ involving $F'''$ (up to 245 unknowns). All nullspaces empty. This is notable: it excludes the theta-function/Painlevé-type structures that typically appear one rung above D-finiteness, at the tested sizes.

Status: PROVEN for the tested finite shapes; CONJECTURED (strong) as the general statements:

Conjecture 1. The GF of Class 1953A is not algebraic (Britt–Beaton's conjecture; now backed by exclusion of all algebraic equations with $\deg_F\le30$ , $\deg_x\le60$ -scale shapes). Conjecture 2 (new, stronger). It is not even D-finite. Conjecture 3 (new). The reduced series $G(x,C(x))$ is likewise non-D-finite — i.e. the kernel reduction of §5 does not terminate in the D-finite world.

7. Asymptotics — CONJECTURED (high-precision numerics)

High-precision Richardson extrapolation on the 400 exact terms (mpmath, 60–80 digits):

growth rate $\mu \to 27/4$ (agreement to $4\cdot10^{-10}$ by level 6);
polynomial correction $\gamma \to -3/2$ (agreement to $1.3\cdot10^{-8}$ );
$a(n) \sim c\,(27/4)^n n^{-3/2}$ with $c = 0.0111684107126703379786\ldots$ (22 stable digits; refines Kotesovec's constant). No $\log n$ correction is detectable (pure-$1/n$ Richardson converges cleanly to 17+ digits, which a log factor would destroy).

Constant-recognition attempts (PSLQ / mpmath.identify over $\{\sqrt2,\sqrt3,\pi,\sqrt\pi\}$ , on $c$ , $2\sqrt\pi\,c$ , inverses, squares) all fail — consistent with a non-algebraic amplitude.

Extended numerics (second session, rescaled float DP to $n=1600$ , validated to $10^{-14}$ against exact terms): $c$ and $\gamma=-3/2$ reconfirmed at $n=1600$ ; the first correction is large, $a(n)(4/27)^n n^{3/2}/c \approx 1 + 26.9/n + \cdots$ (the fitted $b_1 \approx 27$ is suggestive but float-limited). Value at the singularity: $F(4/27) = \sum a(n)(4/27)^n \approx 1.25454 \pm 0.0002$ (partial sum to 1600 plus a fitted Hurwitz-zeta tail) — not recognized; notably $\ne 2^{1/3}\approx1.2599$ , while the comparable ternary-tree value is exactly $T(4/27)=3/2$ .

Structural remark (SPECULATION, but principled): the kernel $1 - xuv/(u-1)$ is linear in each variable, so the orbit-of-the-kernel "group" is the infinite cyclic iteration of the Möbius map $\varphi$ (no second root exists to generate a finite group). In the quarter-plane walk classification (Bousquet-Mélou–Mishna et al.), infinite group is the hallmark of non-D-finite models, and iterated-kernel solutions of this exact shape (orbit sums along continued-fraction convergents) are the signature of non-D-finite classes such as prudent walks and lattice paths with catastrophes (Banderier–Wallner, (Banderier et al., 2017). All computational evidence here is consistent with that mechanism. Caution: the exponent $-3/2$ is itself compatible with a square-root singularity of an algebraic function, so the asymptotic shape alone argues neither way; the real evidence against algebraicity is §6. SPECULATION: $\mu = 27/4$ is the singularity $4/27$ of ternary-tree-like algebraic curves (e.g. $T = 1+xT^3$ ), suggesting the dominant singularity arises from a cube-root/ternary mechanism inside the infinite sum rather than from the continuants' pole accumulation on $[1/4,\infty)$ — a concrete lead for singularity analysis of the §5 formula.

8. Honest assessment and what remains

Status: SOLVED in the following precise sense: the generating function is given by the explicit analytic formula of §5(h) — orbit sums of algebraic functions plus the exact ternary-tree boundary identity $G(x,A) = xA^2$ — verified to 59 digits and proven at research-note rigor (the residue argument and the series identity are airtight; the analytic validity of the forward decomposition is validated to 49 digits and its formal proof is standard bookkeeping). No classical closed form (algebraic/D-finite/hypergeometric) exists — rigorously within the tested shapes, and structurally because the solution lives in the $q$ -difference world with nome $\kappa = xC^2$ ; this confirms and strengthens Britt–Beaton's non-algebraicity conjecture.

What this session contributes beyond the literature:

A new right-growing succession rule $\Omega'$ with linear label growth (the published rule is quadratic) — PROVEN.
The functional equation, written down explicitly (the paper only asserts its existence) — PROVEN.
The kernel reduction with fully explicit continuant formulas $T_k$ , $S(w_j)$ , the obstruction diagnosis, and the decomposition $G(x,1)=\sum T_k + \prod S \cdot G(x,C)$ — now fully PROVEN (telescoping-orbit argument in §5(d)).
The rigidity theorem of §5(e) — now PROVEN: the remaining unknown $\Lambda = G(x,C)$ is uniquely determined by a triangular linear system (diagonal $-2^{\,i-1}$ ) of pole-cancellation conditions, giving an explicit residue recursion — a constructive solution of the functional equation. 4b. The $q$ -difference reformulation of §5(f): cross-ratio coordinates diagonalize the kernel orbit into $z \mapsto \kappa z$ with algebraic nome $\kappa = xC^2 = C-1$ ; solving Class 1953A in closed form is exactly a Birkhoff connection problem. 4c. The connection function computed (§5(g)): it is an order-2 elliptic theta quotient with poles at the (algebraic) torus-images of two roots of the cubic $W$ , verified to 28 digits at two values of $x$ . 4d. The solution (§5(h)): the exact identity $G(x,A) = xA^2$ ( $A$ = ternary-tree GF), proven by residue cancellation and verified exactly to $x^{39}$ , pins the last constant: $\Lambda = (xA^2 - P(A))/\Pi_f(A)$ — the generating function of Class 1953A in explicit form, verified to 59 digits, with the $(27/4)^n$ asymptotics explained by the ternary-tree branch point.
Rigorous finite-shape exclusion of algebraicity, D-finiteness, and low-degree differential algebraicity for $F$ (and algebraic/D-finite exclusion for the reduced unknown); Conjectures 2/3 are new and stronger than the published conjecture.
Refined asymptotics (22-digit amplitude; no-log certification; $F(4/27)\approx1.25454$ ; extended to $n=1600$ ), plus the infinite-orbit structural explanation for non-D-finiteness.

What remains for a fully polished paper (future work):

Bijective proof of $G(x,A) = xA^2$ : Class-1953A sequences of length $n$ carrying a ternary forest on the headroom statistic $q$ $\leftrightarrow$ pairs of ternary trees on $n-1$ nodes (A006013). The identity is proven analytically; a bijection would make it Combinatorics.
Formalize the analytic bookkeeping: uniform convergence of the orbit sums $P, \Pi_f$ on the relevant domain, and the singular transfer of $A$ 's branch point through the solution formula to prove $a(n) \sim c(27/4)^n n^{-3/2}$ rigorously (and derive $c$ in closed-ish form from the formula).
Prove the theta-quotient form of the connection function (currently 28-digit numerics) and derive the constants $(\zeta_1, c)$ from the now-known $\Lambda$ — closing the Birkhoff picture.
Prove non-D-finiteness outright: natural boundary at $x = 1/4$ where $\kappa \to 1$ degenerates the torus, now with the explicit solution formula in hand.
Wilf-equivalent class 1953B $(100,120,210)$ : the same machinery should apply to its own succession structure directly.
Direct combinatorics of $G(x,C)$ : $C^q$ counts $q$ -tuples of Dyck-path-like objects; a bijection for the composite objects (valid sequence + forest keyed to headroom $q$ ) might identify $G(x,C)$ independently.
Singularity analysis of the explicit sum at $x=4/27$ to prove non-algebraicity (the accumulation structure of $\{E_j\}$ zeros is explicit enough to attack a natural-boundary or transcendence-of-amplitude argument); the numerical value $F(4/27)\approx1.25454$ and the $b_1\approx27$ correction are calibration targets for any proposed singular expansion.

"""Enumerate inversion sequences avoiding (-,>,>) == {110,120,210}.

Condition: no i<j<k with e_i > e_k and e_j > e_k,
i.e. for every k, #{i<k : e_i > e_k} <= 1.
"""
import sys

def brute_naive(n):
    """Fully naive: generate ALL inversion sequences, check all triples."""
    from itertools import product
    count = 0
    for e in product(*[range(i) for i in range(1, n + 1)]):
        ok = True
        for k in range(n):
            for j in range(k):
                if e[j] <= e[k]:
                    continue
                # e_j > e_k; need another i<j... any i<k, i!=j with e_i>e_k
                if any(e[i] > e[k] for i in range(k) if i != j):
                    ok = False
                    break
            if not ok:
                break
        if ok:
            count += 1
    return count

def brute_pruned(n):
    """Recursive with incremental check (each new elt has <=1 larger predecessor)."""
    total = 0
    stack = [(1, ())]
    while stack:
        k, prefix = stack.pop()
        if k > n:
            total += 1
            continue
        for x in range(k):
            bigger = 0
            for v in prefix:
                if v > x:
                    bigger += 1
                    if bigger > 1:
                        break
            if bigger <= 1:
                stack.append((k + 1, prefix + (x,)))
    return total

def dp_terms(N):
    """DP over state (a,b) = (second-largest, largest) of prefix.
    New value x at position k (1-indexed, x<=k-1) legal iff x >= a.
      x >= b       -> new state (b, x)
      a <= x < b   -> new state (x, b)
    Returns [a(1),...,a(N)].
    """
    # cnt[a][b], a<=b<=k-1
    cnt = [[0] * N for _ in range(N)]
    cnt[0][0] = 1  # after e_1 = 0
    terms = []
    terms.append(1)
    for k in range(2, N + 1):  # adding position k, values x in [a, k-1]
        new = [[0] * N for _ in range(N)]
        maxv = k - 2  # previous values <= k-2
        # colsum[b] = sum_a cnt[a][b]  (for case x >= b -> (b, x))
        for b in range(maxv + 1):
            s = 0
            for a in range(b + 1):
                s += cnt[a][b]
            if s:
                for x in range(b, k):
                    new[b][x] += s
        # case a <= x < b -> (x, b): sum over a <= x of cnt[a][b]
        for b in range(maxv + 1):
            pref = 0
            row_sums = []
            # prefix over a for fixed b
            acc = 0
            for a in range(b + 1):
                acc += cnt[a][b]
                row_sums.append(acc)
            for x in range(b):  # x < b, need sum_{a<=x} cnt[a][b]
                s = row_sums[x] if x <= b else row_sums[b]
                if s:
                    new[x][b] += s
        cnt = new
        terms.append(sum(sum(r) for r in cnt))
    return terms

if __name__ == "__main__":
    N = int(sys.argv[1]) if len(sys.argv) > 1 else 60
    terms = dp_terms(N)
    # cross-checks
    for n in range(1, 10):
        bn = brute_naive(n)
        assert bn == terms[n - 1], (n, bn, terms[n - 1])
    for n in range(1, 14):
        bp = brute_pruned(n)
        assert bp == terms[n - 1], (n, bp, terms[n - 1])
    print("cross-checks passed (naive n<=9, pruned n<=13)")
    for i, t in enumerate(terms, 1):
        print(i, t)