This note extends the recent work of S. Azizi, A. S. Janfada (2024) on the symmetric treatment of “up” and “down” Steenrod powers for an odd prime p. We give a rigorous proof that their recursively defined Triangular algorithm agrees with the algebraic action of the up Steenrod powers \(\cal{P}^k\) on polynomial algebras, thereby formalizing the harmonic patterns they observed. Building on this, we establish a multivariable extension of their one-variable formula: for a monomial xα and any k ⩾ 0, the Cartan-Lucas factorization yields an explicit expansion of \(\cal{P}^k(x^\alpha)\) whose nonvanishing is governed coordinatewise by the digitwise partial order \(\preceq_p\) . For a general polynomial f, we obtain a support-level description \({\text{supp}}(\cal{P}^k(f))\subseteq\cal{S}_k(f) =\{\beta=\alpha+(p-1)\kappa\colon\alpha\in{\text{supp}}(f),\, |\kappa|=k,\, \kappa_i\preceq_p\alpha_i \},\) together with an explicit coefficient formula. On the combinatorial side, we identify the 0/1 triangular matrices [Up](t) with the Kronecker powers T p ⊗t of the p × p upper-triangular all-ones matrix Tp, proving the digitwise characterization \([U_p](t)_{k,d}=1\iff k\preceq_p d\) . Via graded duality, the same digitwise criterion yields an analogous support-level description for the down Steenrod powers \(\cal{P}_k\) on the divided power algebra DP(n), and we illustrate the resulting row-shift dictionary between up and down patterns by explicit 0/1 heatmaps for p = 3.
