QSymM

The monomial basis $M_\alpha$ of the quasi-symmetric functions, indexed by a composition. The product is the quasi-shuffle, which may ADD two leading parts as well as interleave them.

WikipediaQuasisymmetric functionWikidataQ7269535

QSymM([1,2])

M_\alpha

for the composition

\alpha

NSymH([1,2])the complete basis

H_\alpha

of NSym, whose product is concatenation

Typed signatures

Type	Package	Applies
`(list<integer>) -> number`	`hopf`	always

Domain: Hopf algebras

Details

$M_1 \cdot M_1 = 2M_{1,1} + M_2$ — the $M_2$ term is the overlap, and it is what makes this a QUASI-shuffle rather than an ordinary shuffle
QSym is commutative; NSym is not, which is what its name records
Both are graded with $2^{n-1}$ basis elements in degree $n$ — one per composition of $n$
The two share an index set but are different algebras: mixing $H_\alpha$ and $M_\alpha$ in one product is refused
Use the ordered product (NonCommutativeMultiply or $\otimes$ ); see Coproduct for the other half of the structure