How Are Euler Numbers Linked to Bernoulli Numbers via Ordered Partitions?

Ever wondered why Euler numbers and Bernoulli numbers often appear together in formulas? The answer lies in a surprising combinatorial link. In a recent paper the authors introduced two recursive functions, K_b(n) and K_e(n), that turn ordered partitions of an integer n into coefficients tied to these famous numbers. An ordered partition simply breaks n into a sum of positive numbers, keeping the order. For example, 3 can be written as 1+2, 2+1 or 1+1+1.

The key formulas are:
"B_{2n}= -\frac{(2n)!}{2^{2n}-2}\,K_b(n)" and "E_{2n}= (2n)!\,K_e(n)". In practice, by computing K_b(n) and K_e(n) with their recursions, you directly obtain the values of B_{2n} and E_{2n}. The magic is that K_b(n) and K_e(n) can be expressed as sums over all ordered partitions of n, with alternating signs and factorials of even or odd numbers. This turns an analysis problem (Bernoulli and Euler numbers) into a combinatorial puzzle where each partition adds a piece to the final picture.

The result is more than a mathematical curiosity: it shows how seemingly different structures—power series, integrals, and partition counts—are actually facets of the same gem. For number‑loving minds, it’s an invitation to explore hidden connections between algebra, analysis, and combinatorics, and to see mathematics as a giant building game.

The functions K_b(n) and K_e(n) originate from two very similar recursions, differing only in the parity of the factorials in the denominators: odd factorials for K_b and even factorials for K_e. This mirrors the nature of Bernoulli numbers (linked to even powers) and Euler numbers (associated with odd powers).

Formally, K_b(0)=K_e(0)=1. For n≥1, K_b(n)= -∑_{n'=0}^{n-1} K_b(n')/(2(n-n')+1)!, while K_e(n)= -∑_{n'=0}^{n-1} K_e(n')/(2(n-n'))!. These recursions can be solved via generating functions. Define G_b(x)=∑_{n≥0} K_b(n) x^{2n+1} and G_e(x)=∑_{n≥0} K_e(n) x^{2n}. One proves that G_b(x)= -x/ (e^{x}+e^{-x}) and G_e(x)= -x/ (e^{x}-e^{-x}), directly linking the functions to hyperbolic series.

The next step is the combinatorial translation. An ordered partition λ⊢n is a sequence (b_1,…,b_k) with Σb_i=n. The sign (‑1)^{ℓ(λ)} stems from the alternating nature of the recursion, while the product of factorials in the denominators arises from counting internal permutations within each block of the partition. In other words, each block of size b_i contributes (2b_i+1)! or (2b_i)! depending on whether we deal with K_b or K_e.

Connecting K_b and K_e to Bernoulli and Euler numbers yields:
B_{2n}= -\frac{(2n)!}{2^{2n}-2} K_b(n)\quad and\quad E_{2n}= (2n)! K_e(n).
This identity is elegant because the factor (2^{2n}-2) appears only in the Bernoulli case, reflecting their vanishing for odd indices. Moreover, the presence of (2n)! in both formulas highlights a deep symmetry: Euler numbers are, in a sense, “unnormalized” Bernoulli numbers.

From the viewpoint of generating‑function theory, the identities show that K_b and K_e are coefficients in the expansion of inverse hyperbolic functions, explaining why they naturally arise in contexts involving power‑sum calculations and integrals of trigonometric functions. This connection opens the door to further generalizations, for instance toward Genocchi numbers or Euler polynomials, where ordered partitions continue to provide an effective combinatorial description.