The matrix identity presented below is a specialization of the more general result displayed in the MSE-Q "Lah and associahedra partition polynomials and symmetric functions (reference request)".
I'm looking for combinatorial models of the simple matrix identity given below ($\dagger$) that transforms the factorial-normalized, bivariate Narayana polynomials into the factorial-normalized elementary homogeneous symmetric polynomials in two indeterminates via the refined Lah partition polynomials, reducing the set of bivariate Narayana polynomials to the arithmetic and geometric means of the two indeterminates. (Note the factorial normalization can be achieved by introducing into the formulation diagonal matrices with the factorials along the diagonal.)
The Narayana polynomials (OEIS A001263), the refined Lah partition polynomials (A130561, a.k.a. the elementary Schur polynomials), and the elementary and complete homogeneous symmetric polynomials are encountered frequently in algebraic and geometric combinatorics, number theory, algebra, and analysis. The umbral calculus of Sheffer polynomials provides an easy, short proof of the following matrix identity relating these polynomials:
$(\dagger)$ $$[E] = [L][N] $$
where
$$[N] = (1, -(a+b), 2!(a^2+3ab+b^2), -3!(a^3+6a^2b+6ab^2+b^3), ...)^T$$
is a column vector of signed bivariate Narayana polynomials multiplied by the factorials;
$[L]$ is the lower triangular matrix of coefficients of the row polynomials defined by the e.g.f.
$$\exp\left(x \frac{t}{(1-at)(1-bt)} \right) = \exp(x \cdot tH(t;a,b)) $$
$$ = 1 + x t + (x^2 + 2 (a + b)x) \frac{t^2}{2!}+ (x^3 + 6 (a+ b) x^2 + 6 (a^2 + a b + b^2)x) \frac{t^3}{3!}+ (x^4 + 12 (a + b) x^3 + 12 (3 a^2 + 4 a b + 3 b^2) x^2 + 24 (a^3 + a^2b + ab^2 + b^3)x) \frac{t^4}{4!}+ \cdots$$
with the first few rows being
$1$
$0,1$
$0,2 (a + b), 1$
$0, 6(a^2 + ab + b^2), 6 (a + b), 1$
$0, 24(a^3 + a^2b + ab^2 + b^3), 12 (3 a^2 + 4 ab+ 3 b^2),12 (a + b), 1$
i.e., the refined Lah partition polynomials of A130561 (partially compiled in the Lang link in the OEIS entry) with the indeterminates, for $n \geq 0 $, $x[n+1] = h_n(a,b)\cdot x$ where $h_n(a,b)$ are the complete homogeneous symmetric polynomials $h_n(a,b) = \frac{a^{n+1} -b^{n+1}}{a-b}$ with g.f.
$$H(t;a,b) = 1 + h_1(a,b) t + h_2(a,b) t^2 + \cdots = \frac{1}{(1-at)(1-bt)}$$
$$ = 1 + (a+b)t + 2(a^2+ab+b^2) \frac{t^2}{2!} + 6(a^3+a^2b+ab^2+b^3)\frac{t^3}{3!} + \cdots;$$
and
$$[E] = (1, -(a+b), 2ab, 0, 0, ...)^T$$
is the column vector of associated signed elementary symmetric polynomials multiplied by factorials with e.g.f.
$$E(t;a,b) = (1-at)(1-bt) = 1+(-(a+b))t + 2ab \frac{t^2}{2!} = 1+e_1(a,b)t+2e_2(a,b)\frac{t^2}{2!}.$$
The first few rows of $[L]$ in terms of the $h_n(a,b)$ are
$1$
$0,1$
$0,2 h_1, 1$
$0, 6h_2, 6 h_1, 1$
$0, 24h_3, 12(2h_2+ h_1^2),12 h_1, 1.$
The matrix identity $(\dagger)$ is equivalent to the umbral composition identity
$E_n = L_n(N.)$
where, e.g.,
$E_2 = L_2(x)|_{x^k = (N.)^k = N_k} = (x^2 + 2 (a + b)x) |_{x^k = (N.)^k = N_k} $
$ = (N.)^2 + 2 (a + b)(N.)^1 = N_2 + 2 (a + b)N_1$
$ = (2!(a^2+3ab+b^2)) + 2 (a + b)(-(a+b)) =2ab = 2e_2(a,b) $.
