Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$

Let (n) =[x₁,…,xₙ] be the polynomial algebra in n variables xᵢ, of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on (n) according to well known rules. A major problem in algebraic topology is that of determining ⁺(n), the image of the action of the positively graded part of . We are interested in the related problem of determining a basis for the quotient vector space (n) = (n)/⁺(n). Both (n) =d ≥ 0 ᵈ(n) and (n) are graded, where ᵈ(n) denotes the set of homogeneous polynomials of degree d. In this paper we give explicit formulae for the dimension of (n) in degrees less than or equal to $12.$