Abstract
In the aim of 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^\dagger)^{r_n}a^{s_n}\cdots(a^\dagger)^{r_1}a^{s_1}=(a^\dagger)^\alpha\displaystyle\sum S_{\bf{r,s}}(k)(a^\dagger)^k a^k
, where
\alpha
is assumed to be non-negative. These objects are used to define a combinatorial Hopf algebra which projects to the enveloping algebra of the Heisenberg Lie algebra. Here, we propose a new variant 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 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. The Fusion algebra
\mathcal{F}
defined using formal variables 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 Hopf structure. We recognise two already known combinatorial Hopf subalgebras of
\mathcal{B}
: WSym the algebra of word symmetric functions indexed by set partitions and BWSym the algebra of biword symmetric functions indexed by set partitions into lists.