# fTVector -- compute the ft-vector of a given t-spread ideal of a polynomial ring

## Synopsis

• Usage:
fTVector(I,t)
• Inputs:
• I, a t-spread ideal of polynomial ring
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• a list, a list of nonnegative integers representing the ft-vector of the t-spread ideal I

## Description

Let I be a t-spread ideal of the polynomial ring $S=K[x_1,\ldots,x_n]$ One can define the $f_\texttt{t}$-vector of I as $f_\texttt{t}(\texttt{I})=\left( f_{\texttt{t},-1}(\texttt{I}), f_{\texttt{t},0}(\texttt{I}), \ldots, f_{\texttt{t},j}(\texttt{I}), \ldots \right),$
where $f_{\texttt{t},j-1}(\texttt{I})=|[S_j]_t|-|[I_j]_t|$ and $[I_j]_t$ is the t-spread part of the $j$-th graded component of I.
In an equivalent way, let $\Delta$ be the associated simplicial complex of I. So, $f_{\texttt{t},j}(\texttt{I})$ is the cardinality of the set $\left\{F\ :\ F \text{ is a } j \text{-dimensional face in } \Delta \text{ and } x_F=\prod_{i\in F}{x_i} \text{ is a } t \text{-spread monomial}\right\}.$

Example:

 i1 : S=QQ[x_1..x_8] o1 = S o1 : PolynomialRing i2 : fTVector(ideal {x_1*x_3,x_2*x_5*x_7},1) o2 = {1, 8, 27, 49, 50, 27, 6, 0, 0} o2 : List i3 : fTVector(ideal {x_1*x_3,x_2*x_5*x_7},2) o3 = {1, 8, 20, 15, 2} o3 : List