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In the aim to understand the generalization of Stirling numbers occurring inthe bosonic normal ordering problem, several combinatorial models have beenproposed. In particular, Blasiak \emph{et al.} defined combinatorial objectsallowing to interpret the number of $S_{\bf{r,s}}(k)$ appearing in the identity$(a^\dag)^{r_n}a^{s_n}\cdots(a^\dag)^{r_1}a^{s_1}=(a^\dag)^\alpha\displaystyle\sumS_{\bf{r,s}}(k)(a^\dag)^k a^k$, where $\alpha$ is assumed to be non-negative.These objects are used to define a combinatorial Hopf algebra which specializesto the enveloping algebra of the Heisenberg Lie algebra. Here, we propose a newvariant of this construction which admits a realization with variables. Thismeans that we construct our algebra from a free algebra $\mathbb{C}\langle A\rangle$ using quotient and shifted product. The combinatorial objects(B-diagrams) are slightly different from those proposed by Blasiak \emph{etal.}, but give also a combinatorial interpretation of the generalized Stirlingnumbers together with a combinatorial Hopf algebra related to Heisenberg Liealgebra. The main difference comes from the fact that the B-diagrams have thesame number of inputs and outputs. After studying the combinatorics and theenumeration of B-diagrams, we propose two constructions of algebras calledFusion algebra $\mathcal{F}$, defined using formal variable and another algebra$\mathcal{B}$ constructed directly from the B-diagrams. We show the connectionbetween these two algebras and that $\mathcal{B}$ can be endowed with a Hopfstructure. We recognize two already known combinatorial Hopf subalgebras of$\mathcal{B}$ : $\mathrm{WSym}$ the algebra of word symmetric functions indexedby set partitions and $\mathrm{BWSym}$ the algebra of biword symmetricfunctions indexed by set partitions into lists.

A combinatorial Hopf algebra for the boson normal ordering problem