On the Steenrod length of real projective spaces: finding longest chains in certain directed graphs Original Research Article

We consider a problem in homotopy theory that has been reduced to enumerative combinatorics by Christensen in (Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta, Ph.D. Thesis, MIT, Cambridge, MA). (Although the motivation is topological, the problem is of independent combinatorial interest.) Consider a directed graph with vertices labeled by the non-negative integers. There is an outgoing edge from vertex n for each 0 in the binary representation of n (excluding ‘leading zeros’); if the 2s term in n is 0, then the corresponding edge goes from vertex n to vertex n − 2s. Let f(n) be the length of the longest sequence of edges starting with vertex n, and let g(n) be the longest sequence of edges starting from any vertex q ⩽ n. Then we analyze several unusual properties of f, and prove that the frequency table of g(n) is a sequence of non-decreasing powers of 2, where 2a appears a + 1 times (a ⩾ 1).