Daisy cubes and distance cube polynomial

Let X $\subseteq$ {0, 1} n. Then the daisy cube Q n (X) is introduced as the sub-graph of Q n induced by the intersection of the intervals I(x, 0 n) over all x $\in$ X. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If u is a vertex of a graph G, then the distance cube polynomial D G,u (x, y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Q k at a given distance from the vertex u. It is proved that if G is a daisy cube, then D G,0 n (x, y) = C G (x + y -- 1), where C G (x) is the previously investigated cube polynomial of G. It is also proved that if G is a daisy cube, then D G,u (x, --x) = 1 holds for every vertex u in G.