arXiv2016
In the aim to understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak \emph{et al.} defined combinatorial objects allowing to interpret the number of $S_{\bf{r,s}}(k)$ appearing in the identity $(a^†)^{r_n}a^{s_n}\cdots(a^†)^{r_1}a^{s_1}=(a^†)^α\displaystyle\sum S_{\bf{r,s}}(k)(a^†)^k a^k$, where $α$ is assumed to be non-negative. These objects are used to define a combinatorial Hopf algebra which specializes to the enveloping algebra of the Heisenberg Lie algebra. Here, we propose a new variant of this construction which admits a realization with variables. This means 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{et al.}, but give also a combinatorial interpretation of the generalized Stirling numbers together with a combinatorial Hopf algebra related to Heisenberg Lie algebra. The main difference comes from the fact that the B-diagrams have the same number of inputs and outputs. After studying the combinatorics and the enumeration of B-diagrams, we propose two constructions of algebras called Fusion algebra $\mathcal{F}$, defined using formal variable and another algebra $\mathcal{B}$ constructed directly from the B-diagrams. We show the connection between these two algebras and that $\mathcal{B}$ can be endowed with a Hopf structure. We recognize two already known combinatorial Hopf subalgebras of $\mathcal{B}$ : $\mathrm{WSym}$ the algebra of word symmetric functions indexed by set partitions and $\mathrm{BWSym}$ the algebra of biword symmetric functions indexed by set partitions into lists.