A Solomon Mackey formula for graded bialgebras

Given a graded bialgebra $H$, we let $\Delta^{\left[ k\right] }:H\rightarrow H^{\otimes k}$ and $m^{\left[ k\right] }:H^{\otimes k}\rightarrow H$ be its iterated (co)multiplications for all $k\in\mathbb{N}$. For any $k$-tuple $\alpha=\left( \alpha_{1},\alpha_{2},\ldots,\alpha_{k}\right) \in\mathbb{N}^{k}$ of nonnegative integers, and any permutation $\sigma$ of $\left\{ 1,2,\ldots,k\right\}$, we consider the map $p_{\alpha,\sigma}:=m^{\left[ k\right] }\circ P_{\alpha}\circ\sigma^{-1}\circ\Delta^{\left[ k\right] }:H\rightarrow H$, where $P_{\alpha}$ denotes the projection of $H^{\otimes k}$ onto its multigraded component $H_{\alpha_{1}}\otimes H_{\alpha_{2}}\otimes\cdots\otimes H_{\alpha_{k}}$, and where $\sigma^{-1}:H\rightarrow H$ permutes the tensor factors. We prove formulas for the composition $p_{\alpha,\sigma}\circ p_{\beta,\tau}$ and the convolution $p_{\alpha,\sigma}\star p_{\beta,\tau}$ of two such maps. When $H$ is cocommutative, these generalize Patras's 1994 results (which, in turn, generalize Solomon's Mackey formula). We also construct a combinatorial Hopf algebra $\operatorname*{PNSym}$ ("permuted noncommutative symmetric functions") that governs the maps $p_{\alpha,\sigma}$ for arbitrary connected graded bialgebras $H$ in the same way as the well-known $\operatorname*{NSym}$ governs them in the cocommutative case. We end by outlining an application to checking identities for connected graded Hopf algebras.