Three Infinite Families Separating Schur Positivity, the Strongly Nice Property, and the Nice Property

Abstract

For a graph $G$, Schur positivity of $X_G$ implies that $G$ is strongly nice, and every strongly nice graph is nice. We construct three infinite families separating these properties. We first give a connected family $F_t$, $t\ge6$, that is strongly nice but not Schur positive. We then prove that homogeneous strongly nice symmetric functions with nonnegative monomial coefficients are closed under multiplication, and hence that strongly nice graphs are closed under disjoint union. As an application, for $H=K_{3,3}-e$, the graphs \[ M_t=H\sqcup K_t,\qquad t\ge3, \] form a disconnected family that is strongly nice but not Schur positive. Finally, we define \[ N_r=K_r\vee(K_2\sqcup2K_1),\qquad r\ge2, \] and prove that every $N_r$ is connected and nice but not strongly nice. We also introduce the level-$k$ nice property and show that the level depth of $N_r$ is $4r!$.