I've identified the generating functions for the tangent Chern numbers of the complex projective spaces $CP^n$ given in "Algebraic topology of the Lagrange inversion" by Victor Buchstaber and Alexander Veselov as
$$C^{\tau}(CP^n) = (n+1) \cdot N_{n+1}(t_1,t_2,...,t_n)$$
where $N_n(t_1,t_2,...,t_n)$ are the partition polynomials of OEIS A134264, known as the refined Narayana polynomials, the refined h-polynomials of the associahedra, the noncrossing partitions polynomials of Kreweras, and the Voiculescu polynomials of free probabliity theory, with the first few being
$N_2(t_1) = t_1,$
$N_3(t_1,t_2) = t_1^2 + t_2,$
$N_4(t_1,t_2,t_3) = t_1^3 + 3t_1t_2 + t_3 ,$
$N_5(t_1,t_2,t_3,t_4) = t_1^4 + 4 t_1 t_3 + 6 t_1^2 t_2 + 2 t_2^2 + t_4 .$
The coefficients of $(n+1) \cdot N_{n+1}(t_1,t_2,...,t_n)$ also correspond to reductions of the monomial symmetric polynomials $m_{\lambda \vdash n}^{(n+1)}$ in $(n+1)$ distinct symbols / indeterminates / variables associated with integer partitions of $n$.
For example, for $n = 4$,
$m_{1,1,1,1}^{(5)} = x_1 x_2 x_3 x_4 + x_1 x_2 x_3 x_5 + x_1 x_2 x_4 x_5 + x_1 x_3 x_4 x_5 + x_2 x_3 x_4 x_5$
and
$m_{4}^{(5)} =x_1^4 + x_2^4 + x_3^4 +x_4^4 + x_5^4$
can be associated respectively to $5t_1 t_1 t_1 t_1 t_1 = 5 t_1^4$ and $5t_4$ by erasing the subscripts and lowering the exponents with the number of summands being $5$.
With the same umbral transformations,
$m_{1,1,1,1}^{(5)} + m_{1,3}^{(5)} + m_{1,1,2}^{(5)} + m_{2,2}^{(5)} + m_{4}^{(5)}$
becomes
$5t_1^4 + 20 t_1 t_3 + 30 t_1^2 t_2 + 10 t_2^2 + 5t_4 = 5 \cdot N_5(t_1,t_2,t_3,t_4)$
with the coefficients corresponding to the number of summands in the associated symmetric monomials.
Question
What other significant algebraic or topological interpretations do $C^{\tau}(CP^n) = (n+1) \cdot N_{n+1}(t_1,t_2,...,t_n)$ have?
(These are refinements of the polynomials associated with the Grand Dyck paths of A118963--let all $t_k = x$ to obtain simple multiples of the Narayana triangle A001263.)
