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countTLexMon -- give the cardinality of the smallest initial t-lex segment containing a given monomial



the function countTLexMon(u,t) gives the cardinality of $L_t\{u\}$, the initial t-lex segment defined by u, that is, the number of all the t-spread monomials greater than u, with respect to $>_\mathrm{slex}.$
If FixedMax is true the function countTLexMon(u,t,FixedMax=>true) gives the number of all the t-spread monomials having maximum of the support equal to $\max\textrm{supp}(u)$ and greater than u, with respect to $>_\mathrm{slex}.$
Given a $t$-spread monomial $u=x_{i_1}x_{i_2}\cdots x_{i_d}$, we define $\textrm{supp}(u)=\{i_1,i_2,\ldots, i_d\}$.
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.


i1 : S=QQ[x_1..x_9]

o1 = S

o1 : PolynomialRing
i2 : countTLexMon(x_2*x_5*x_8,2)

o2 = 21
i3 : countTLexMon(x_2*x_5*x_8,3)

o3 = 7

See also

Ways to use countTLexMon :

For the programmer

The object countTLexMon is a method function with options.