Combinatorics of multiboundary singularities \(B_n^l\) and the Bernoulli-Euler numbers (Q1812407)

The paper deals with the families of polynomials \[ x^n+ \lambda_2x^{n-1}+ \cdots + \lambda_{n-1}x \] (where \(x\) is a variable and \(\lambda's\) are parameters), defined on the real line with \(l\) distinguished points \(b_1, \dots ,b_l\). A pair -- a polynomial and a set of \(b_i\)'s -- is called a very nice M-Morsification of a multiboundary singularity \(B_n^l\) if all critical points of the polynomial are real and nondegenerated and all critical values and values at the distinguished points are pairwise distinct (i.e. together there are \(n-1+l\) distinct values). The set of parameters \((\lambda, b)\) corresponding to the very nice M-Morsifications is open in \(\mathbb{R}^{n-2} \times \mathbb{R}^l\). The goal of the paper is to determine the number of its connected components \(K^l_n\). The author claims that \[ K^{l+1}_{n-2}=K^l_n -nlK^{l-1}_n,\;\; \text{ for } l \geq 1, \] and \(K^0_n=K_{n-1}\), \(K^1_n=K_{n+1}\), where \(K_n\) are Bernoulli-Euler numbers.