  Based on the covariant color-kinematics duality, we investigate combinatorial
and algebraic structures underlying their Bern-Carrasco-Johansson (BCJ)
numerators of tree-level amplitudes in Yang-Mills-scalar (YMS) theory. The
closed-formulae for BCJ numerators of YMS amplitudes and the pure-Yang-Mills
(YM) ones exhibit nice quasi-shuffle Hopf algebra structures, and interestingly
they can be viewed as summing over boundaries of all dimensions of a
combinatorial permutohedron. In particular, the numerator with two scalars and
$n{-}2$ gluons contains Fubini number ( $F_{n{-}2}$ ) of terms in one-to-one
correspondence with boundaries of a $(n{-}3)$-dimensional permutohedron, and
each of them has its own spurious-pole structures and a gauge-invariant
numerator (both depending on reference momenta). From such Hopf algebra or
permutohedron structure, we derive new recursion relations for the numerators
and intriguing "factorization" on each spurious pole/facet of the
permutohedron. Similar results hold for general YMS numerators and the pure-YM
ones. Finally, with a special choice of reference momenta, our results imply
BCJ numerators in a heavy-mass effective field theory with two massive
particles and $n{-}2$ gluons/gravitons: we observe highly nontrivial
cancellations in the heavy-mass limit, leading to new formulae for the
effective numerators which resemble those obtained in recent works.
