Betti numbers of powers of path ideals of cycles

Let J_n,m = (x_1⋯ x_m,x_2 ⋯ x_m+1,…,x_nx_1⋯ x_m-1) be the m-path ideal of a cycle of length n ≥ 5 over a polynomial ring S = k[x_1,…,x_n]. Let t≥ 1 be an integer. We show that J_n,m^t has a linear free resolution and give a precise formula for all of its Betti numbers when m = n-1, n-2.