Linear cumulant conjecture for colored interlacing triangles

Proof of the all-order linear-cumulant conjecture. Let

$$P_n(q)=2^{1-n}T_2(n;q)=\sum_{r\ge0}a_r(n)q^r, \qquad \kappa_r(n)=r![q^r]\log P_n(q).$$

We prove, for every $r\ge1$, that $\kappa_r(n)$ is exactly affine in $n$ whenever $n\ge5r$.

The $n-1$ independent top-row interface involutions from Corollary 2.3 and Proposition 3.4(iii) of the source act freely and preserve the energy $\psi$. Thus $P_n(q)$ enumerates their orbits, represented by sorting each interface pair. Cutting a canonical configuration at every ordered direct-sum boundary gives its unique sequence of irreducible configurations. Energy is additive under ordered direct sum. Therefore, with $C(z,q)$ the generating function of nonempty irreducible configurations,

$$F(z,q):=\sum_{n\ge0}P_n(q)z^n=\frac1{1-C(z,q)}. \tag{1}$$

Here is the finite-energy locality estimate that controls the irreducibles. Let $b=(b_1,\ldots,b_n)$ be the bottom permutation, and let $A_j$ denote the active colors immediately before its $j$-th entry. Interlacing gives $|A_j|=j$ and $b_j\in A_j$. By source equation (3.6),

$$e_j=\#\{c\in A_j:c>b_j\}, \qquad \psi=\sum_{j=1}^n e_j.$$

Define

$$h_i=\#\{j\le i:b_j>i\}, \quad D=\sum_{i=1}^{n-1}h_i =\sum_j(j-b_j)_+ =\sum_j(b_j-j)_+, \quad r_j=\#(A_j\setminus[j]).$$

Since $(j-b_j)_+\le e_j$,

$$D\le\psi, \qquad r_j\le e_j+(b_j-j)_+, \qquad \sum_j r_j\le2\psi. \tag{2}$$

The following sufficient cut criterion avoids any assumptions about intermediate active-set healing:

$$h_i=r_i=r_{i+1}=0 \quad\Longrightarrow\quad i\text{ is an ordered direct-sum cut}. \tag{3}$$

Indeed $h_i=0$ makes the first $i$ bottom colors exactly $[i]$, while $r_i=r_{i+1}=0$ gives $A_i=[i]$ and $A_{i+1}=[i+1]$. The interface pair must therefore consist of $b_i$ and $i+1$, in canonical increasing order, separating the two direct-sum blocks.

Every noncut is consequently either an interface with $h_i>0$, or is incident to some position with $r_j>0$. Each position is incident to at most two interfaces. Hence

$$\#\{\text{noncut interfaces}\} \le D+2\sum_jr_j \le5\psi.$$

An irreducible configuration of size $n$ has $n-1$ noncuts, and therefore

$$n\le5\psi+1. \tag{4}$$

It follows that

$$C(z,q)=z+\sum_{r\ge1}c_r(z)q^r, \qquad c_r(z)\in\mathbb Z[z], \qquad \deg_z c_r\le5r+1. \tag{5}$$

Since $C(z,0)=z$, there is a unique formal series $R(q)\in1+q\mathbb Q[[q]]$ satisfying

$$C(R(q),q)=1.$$

Fix $K$ and work modulo $q^{K+1}$. Formal division at this root gives

$$1-C(z,q)=\left(1-\frac z{R(q)}\right)H_K(z,q), \qquad H_K(z,0)=1,$$

with $\deg_z[q^j]H_K^{-1}\le5j$. Therefore, for $n\ge5K$, equation (1) yields

$$P_n(q) \equiv \frac{R(q)^{-(n+1)}}{\partial_z C(R(q),q)} \pmod{q^{K+1}}.$$

Taking formal logarithms and setting $K=r$ proves the explicit all-order formula

$$\boxed{\displaystyle \kappa_r(n) = -r!\left( (n+1)[q^r]\log R(q) +[q^r]\log \partial_zC(R(q),q) \right), \qquad n\ge5r. }$$

Its right-hand side is affine in $n$, proving Conjecture 4.5 at every order.

As additional consequences, $a_r(n)$ is eventually a degree-$r$ polynomial, and $r!a_r(n)\in\mathbb Z[n]$. Indeed source Proposition 3.5 gives

$$c_1(z)=z^2+3z^3+z^4, \qquad c_1(1)=5, \qquad R'(0)=-5,$$

so

$$a_r(n)=\frac{5^r}{r!}n^r+O(n^{r-1}).$$

These consequences establish the general eventual-polynomiality and leading-coefficient assertions. No claim is made here about the four separate sharper onset formulas in Conjecture 4.2.

Source: Blitvić and Petrov, Colored interlacing triangles and Genocchi medians, Conjecture 4.5, https://arxiv.org/abs/2602.04390.