On the cohomology of the mod p Steenrod algebra

Let p be an odd prime greater than seven and A the mod p Steenrod algebra. In this paper we prove that in the cohomology of A the product h_1 h_n \tilde \delta _{s + 4}\in {\rm Ext}_A^{s + 6, t(s,n) + s} ({\bf Z}_p , {\bf Z}_p) is nontrivial for n \geq 5, and trivial for n=3, 4, where \tilde \delta _{s + 4} is actually \tilde \alpha _{s+4}^{(4)} described by X. Wang and Q. Zheng, 0 \leq s < p - 4, t(s,n) = 2(p-1)[(s + 1) + (s + 3)p + (s + 3)p^2 + (s + 4)p^3 + p^n ]. We show our results by explicit combinatorial analysis of the (modified) May spectral sequence. The method of proof is very elementary.