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Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N \geq k$.Let $S$ be a subset of ${1,2,...,k-1}$. Define the fundamental polynomial in $N$ variables as $F_{k,S}(x_1,x_2,...,x_N):= \sum\limits_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq N \ ; \ j \in S \implies i_j < i_{j+1}}^{}{x_{i_1} x_{i_2} \cdots x_{i_k}}$.

Question: Is there an existing command to obtain those polynomials for a given triple $(N,k,S)$ via Sage?

I found a section about quasi-symmetric functions in Sage but it uses the language of Hopf algebras and I am not sure whether this contains the fundamental polynomials already in this form.

Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N \geq k$.Let $S$ be a subset of ${1,2,...,k-1}$. Define the fundamental polynomial in $N$ variables as $F_{k,S}(x_1,x_2,...,x_N):= \sum\limits_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq N \ ; \ j \in S \implies i_j < i_{j+1}}^{}{x_{i_1} x_{i_2} \cdots x_{i_k}}$.

Question: Is there an existing command to obtain those polynomials for a given triple $(N,k,S)$ via Sage?

I found a section about quasi-symmetric functions in Sage but it uses the language of Hopf algebras and I am not sure whether this contains the fundamental polynomials already in this form.

Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N \geq k$.Let $S$ be a subset of the set with elements ${1,2,...,k-1}$. Define the fundamental polynomial in $N$ variables as $F_{k,S}(x_1,x_2,...,x_N):= \sum\limits_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq N \ ; \ j \in S \implies i_j < i_{j+1}}^{}{x_{i_1} x_{i_2} \cdots x_{i_k}}$.

For example we have $F_{k,\emptyset}=h_k$, the complete symmetric function and $F_{k,{1,...,k-1 }}=e_k$, the elementary symmetric function.

Question: Is there an existing command to obtain those polynomials for a given triple $(N,k,S)$ via Sage?

I found a section about quasi-symmetric functions in Sage but it uses the language of Hopf algebras and I am not sure whether this contains the fundamental polynomials already in this form.

Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N \geq k$.Let $S$ be a subset of the set with elements ${1,2,...,k-1}$. {1,2,...,k-1}. Define the fundamental polynomial in $N$ variables as $F_{k,S}(x_1,x_2,...,x_N):= \sum\limits_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq N \ ; \ j \in S \implies i_j < i_{j+1}}^{}{x_{i_1} x_{i_2} \cdots x_{i_k}}$.

For example we have $F_{k,\emptyset}=h_k$, the complete symmetric function and $F_{k,{1,...,k-1 }}=e_k$, the elementary symmetric function.

Question: Is there an existing command to obtain those polynomials for a given triple $(N,k,S)$ via Sage?

I found a section about quasi-symmetric functions in Sage but it uses the language of Hopf algebras and I am not sure whether this contains the fundamental polynomials already in this form.

Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N \geq k$.Let $S$ be a subset of the set with elements {1,2,...,k-1}. Define the fundamental polynomial in $N$ variables as $F_{k,S}(x_1,x_2,...,x_N):= \sum\limits_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq N \ ; \ j \in S \implies i_j < i_{j+1}}^{}{x_{i_1} x_{i_2} \cdots x_{i_k}}$.

For example we have $F_{k,\emptyset}=h_k$, the complete symmetric function and $F_{k,{1,...,k-1 }}=e_k$, $F_{k,S}=e_k$ for S={1,...,k-1}, the elementary symmetric function.

Question: Is there an existing command to obtain those polynomials for a given triple $(N,k,S)$ via Sage?

I found a section about quasi-symmetric functions in Sage but it uses the language of Hopf algebras and I am not sure whether this contains the fundamental polynomials already in this form.

Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N \geq k$.Let $S$ be a subset of the set {1,2,...,k-1}. Define the fundamental polynomial in $N$ variables as $F_{k,S}(x_1,x_2,...,x_N):= \sum\limits_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq N \ ; \ j \in S \implies i_j < i_{j+1}}^{}{x_{i_1} x_{i_2} \cdots x_{i_k}}$.

For example we have $F_{k,\emptyset}=h_k$, the complete symmetric function and $F_{k,S}=e_k$ for S={1,...,k-1}, the elementary symmetric function.

Question: Is there an existing command or another easy way to obtain those polynomials for a given triple $(N,k,S)$ via Sage?

I found a section about quasi-symmetric functions in Sage but it uses the language of Hopf algebras and I am not sure whether this contains the fundamental polynomials already in this form.