Abstract

In [4], the authors show that if <TEX>${\mathbb{X}}$</TEX> is a <TEX>${\mathbb{k}}-configuration$</TEX> in <TEX>${\mathbb{P}}^2$</TEX> of type (<TEX>$d_1$</TEX>, <TEX>${\ldots}$</TEX>, <TEX>$d_s$</TEX>) with <TEX>$d_s$</TEX> > <TEX>$s{\geq}2$</TEX>, then <TEX>${\Delta}H_{m{\mathbb{X}}}(md_s-1)$</TEX> is the number of lines containing exactly <TEX>$d_s-points$</TEX> of <TEX>${\mathbb{X}}$</TEX> for <TEX>$m{\geq}2$</TEX>. They also show that if <TEX>${\mathbb{X}}$</TEX> is a <TEX>${\mathbb{k}}-configuration$</TEX> in <TEX>${\mathbb{P}}^2$</TEX> of type (1, 2, <TEX>${\ldots}$</TEX>, s) with <TEX>$s{\geq}2$</TEX>, then <TEX>${\Delta}H_{m{\mathbb{X}}}(m{\mathbb{X}}-1)$</TEX> is the number of lines containing exactly s-points in <TEX>${\mathbb{X}}$</TEX> for <TEX>$m{\geq}s+1$</TEX>. In this paper, we explore a standard <TEX>${\mathbb{k}}-configuration$</TEX> in <TEX>${\mathbb{P}}^2$</TEX> and find that if <TEX>${\mathbb{X}}$</TEX> is a standard <TEX>${\mathbb{k}}-configuration$</TEX> in <TEX>${\mathbb{P}}^2$</TEX> of type (1, 2, <TEX>${\ldots}$</TEX>, s) with <TEX>$s{\geq}2$</TEX>, then <TEX>${\Delta}H_{m{\mathbb{X}}}(m{\mathbb{X}}-1)=3$</TEX>, which is the number of lines containing exactly s-points in <TEX>${\mathbb{X}}$</TEX> for <TEX>$m{\geq}2$</TEX> instead of <TEX>$m{\geq}s+1$</TEX>.