The expansion conjecture for the generating functions $T_{a,k}(x)$

Let $\mathcal{S}_{n,k}^{a\prec n}(1324)$ denote the set of $1324$-avoiding permutations in $\mathcal{S}_n$ whose relevant positional parameters are indexed by $a$ and $k$. Define

$$T_{a,k}(x)=\sum\limits_{n=k+1}^{\infty}|\mathcal{S}_{n,k}^{a\prec n}(1324)|x^n,$$

and

$$g_a(x,t)=\sum\limits_{k=1}^{\infty}t^kT_{a,k}(x).$$

Let $T_{1,0}(x)=x$ and $T_{a,0}(x)=|\mathcal{S}_{a-1}(1324)|x^a$. Also, let $f(x)$ denote the generating function introduced earlier in the paper. For $k\geq a$, the following formulas are equivalent:

Expansion conjecture.

$$\sum\limits_{j=0}^k(-1)^j\binom{k}{j}f(x)^jT_{a,k-j}(x)=0.$$

Equivalently,

$$T_{a,k}(x)=\sum\limits_{j=0}^{a-1}\binom{k}{j}f(x)^{k-j}\sum\limits_{i=0}^{j}(-1)^i\binom{j}{i}f(x)^iT_{a,j-i}(x),$$

and

$$T_{a,k}(x)=\sum\limits_{j=0}^{a-1}(-1)^{a-j-1}\binom{k}{j}\binom{k-j-1}{a-j-1}f(x)^{k-j}T_{a,j}(x).$$

The conjecture proposes an explicit relation between $T_{a,k}(x)$ and powers of $f(x)$ together with the lower-indexed functions $T_{a,j}(x)$. It is motivated by the structural relation between $1324$-avoiding permutations in which entries less than $a$ lie to the right of $n$ and the corresponding reduced permutations, as well as by the proof of the earlier theorem for $T_{2,k}(x)$.