# countTStronglyStableMon -- give the cardinality of the t-strongly stable set generated by a given monomial

## Synopsis

• Usage:
countTStronglyStableMon(u,t)
• Inputs:
• u, a t-spread monomial of a polynomial ring
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• an integer, the number of all the t-spread monomials of the t-strongly stable set generated by u

## Description

the function countTStronglyStableMon(u,t) gives the cardinality of $B_t\{u\}$, the t-strongly stable set generated by u, that is, the number of all the t-spread monomials belonging to the smallest t-strongly stable set containing u
This method is not constructive. It uses a theoretical result to obtain the cardinality as the sum of suitable binomial coefficients. The procedure only concerns $\textrm{supp}(\texttt{u}),$ that is, the set $\{i_1,i_2,\ldots, i_d\}$, when $u=x_{i_1}x_{i_2}\cdots x_{i_d}$ is a $t$-spread monomial.
Moreover, a subset $N\subset M_{n,d,t}$ is called a t-strongly stable set if taking a t-spread monomial $u\in N$, for all $j\in \mathrm{supp}(u)$ and all $i,\ 1\leq i\leq j$, such that $x_i(u/x_j)$ is a t-spread monomial, then it follows that $x_i(u/x_j)\in N$.

Examples:

 i1 : S=QQ[x_1..x_9] o1 = S o1 : PolynomialRing i2 : countTStronglyStableMon(x_2*x_5*x_8,2) o2 = 14 i3 : countTStronglyStableMon(x_2*x_5*x_8,3) o3 = 4