We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function methodology of
modern enumerative combinatorics and their applications to computational problems.
We construct a three-parameter deformation of the Hopf algebra $LDIAG$. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the {em product formula} in a simplified version of Quantum Field Theory. This new algebra i
s a true Hopf deformation which reduces to $LDIAG$ for some parameter values and to the algebra of Matrix Quasi-Symmetric Functions ($MQS$) for others, and thus relates $LDIAG$ to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler-Zagier sums.
