Gröbner Bases and Dimension Formulas for Ternary Partially Associative Operads

Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger’s algorithm for Grobner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation \(({*}{*}{*})\), we compute a Grobner 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 \(({*}{*}{*})\) may be even or odd. We use the Grobner bases to calculate the dimension formulas for these operads.