Betti Tables and the Licci Property

We show that the licci property of an ideal cannot, in general, be determined from its Betti table by exhibiting two $\mathfrak{m}$-primary monomial ideals containing the cubes the variables that have identical Betti tables, yet only one is licci.

In contrast, we show that for monomial ideals containing the square of each variable, the licci property is completely determined by the Betti table; indeed, it is determined by its first two columns.