The Relationship Between Euler Numbers and Bernoulli Numbers with Ordered Partitions

In this paper, for every $n\in \mathbb{N}$, the following relationships between the functions $K_b(n)$ and $K_e(n)$ and the Bernoulli and Euler numbers are proved: $$B_{2n}=-\frac{(2n)!}{2^{2n}-2}{K}_b(n),E_{2n}=(2n)!K_e(n)\mathrm{.}$$ The functions $K_b$ and $K_e$ are defined recursively by $$K_b(0)=K_e(0)=1,$$ $$K_b(n)=-\sum _{n^{\mathrm{\prime }}=0}^{n-1}\frac{K_b(n^{\mathrm{\prime }})}{(2(n-n^{\mathrm{\prime }})+1)!},n\ge 1,$$ $$K_e(n)=-\sum _{n^{\mathrm{\prime }}=0}^{n-1}\frac{K_e(n^{\mathrm{\prime }})}{(2(n-n^{\mathrm{\prime }}))!},n\ge 1.$$ Furthermore, we present combinatorial interpretations of these functions in terms of ordered partitions of $n$: $$K_b(n)=\sum _{\lambda \models n}\frac{(-1{)}^{\mathrm{\ell }(\lambda )}}{{\displaystyle \prod _{i=1}^{\mathrm{\ell }(\lambda )}(2b_i+1)!}},n\ge 1,$$ $$K_e(n)=\sum _{\lambda \models n}\frac{(-1{)}^{\mathrm{\ell }(\lambda )}}{{\displaystyle \prod _{i=1}^{\mathrm{\ell }(\lambda )}(2b_i)!}},n\ge 1,$$ where $\lambda =(b_1,b_2,\dots ,b_k)\models n$ and $\mathrm{\ell }(\lambda )=k$.