Description
Let $\texttt{S}=K[x_1,\ldots,x_n]$, $\texttt{t}\geq 1$ and let $\texttt{f}=(f(0),f(1),\ldots,f(d),\ldots)$ be a sequence of nonnegative integers. The sequence
f is the $\texttt{f}_\texttt{t}$-vector of a
t-spread ideal of
S if the following conditions hold:
$f(0)\leq 1,\ f(1)\leq n \text{ and } f(d+1)\leq \texttt{tMacaulayExpansion(f(d),n,d,t,Shift=>true)}$ for all $d>1.$
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.
Example:
i1 : S=QQ[x_1..x_8]
o1 = S
o1 : PolynomialRing
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i2 : f={1,8,20,10,0}
o2 = {1, 8, 20, 10, 0}
o2 : List
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i3 : isFTVector(S,f,2)
o3 = true
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i4 : S=QQ[x_1..x_7]
o4 = S
o4 : PolynomialRing
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i5 : isFTVector(S,f,2)
o5 = false
|