Conjectured formulas for upper-degree degeneracies (co-dimension 1 and 2, and extremal degeneracies)

Prove that the explicit degeneracy formulas for (n−1)-simplices and (n−2)-simplices, and for the largest and second-largest d-degrees, hold for all n. Specifically, establish D^{(n-1)}(0,n)=\tfrac{(n−1)(n+2)}{2}, D^{(n-1)}(1,n)=n+1; D^{(n-2)}(0,n)=\tfrac{(n−2)(n−1)n(3n+11)}{24}, D^{(n-2)}(1,n)=\tfrac{n(n^2+n−4)}{2}, D^{(n-2)}(2,n)=\tfrac{(n−1)n}{2}, D^{(n-2)}(3,n)=n; and for general d, D^{(d)}(M^{(d)}(n),n)=d+2 (n≥1) and D^{(d)}(\widetilde M^{(d)}(n),n)=\tfrac{(d+1)(d+2)}{2} (n≥d+2).