Asymptotics of the number of 2-threshold functions

Elena Zamaraeva, Jovisa D. Zunic

Abstract: A k-threshold function on a rectangular grid of size m×n<math><mi is="true">m</mi><mo is="true">×</mo><mi is="true">n</mi></math> is the conjunction of k threshold functions on the same domain. In this paper, we focus on the case k=2<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn></math> and show that the number of two-dimensional 2-threshold functions is 2512π4m4n4+o(m4n4)<math><mfrac is="true"><mrow is="true"><mn is="true">25</mn></mrow><mrow is="true"><mn is="true">12</mn><msup is="true"><mrow is="true"><mi is="true">π</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup></mrow></mfrac><msup is="true"><mrow is="true"><mi is="true">m</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">o</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">m</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math>.