Gr\"obner bases and dimension formulas for ternary partially associative operads

Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger’s algorithm for Gr"obner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation $(\ast \ast \ast )$, we compute a Gr"obner basis for the ideal generated by partial associativity $((abc)de)+(a(bcd)e)+(ab(cde)$. In the category of $\mathbb{Z}$-graded vector spaces with Koszul signs, the (homological) degree of $(\ast \ast \ast )$ may be even or odd. We use the Gr"obner bases to calculate the dimension formulas for these operads.