We introduce the cyclic major index of a permutation and give bivariate analogue enumerative formula for shuffles with given descent number due to Adin, Gessel, Reiner Roichman, which can be viewed as Stanley's shuffling theorem. This gives an answer question has been posed by Domagalski, Liang, Minnich, Sagan, Schmidt Sietsema again.

One of the most famous results Dedekind is transformation law $$\log \Delta (z)$$ . After a half-century, Rademacher modified Dedekind’s result and introduced an $${{\,\textrm{SL}\,}}_2({\mathbb {Z}})$$ -conjugacy class invariant (integer-valued) function $$\Psi (\gamma )$$ called symbol. Inspired by Ghys’ work on modular knots, Duke–Imamo?lu–Tóth (2017) constructed hyperbolic analogue In this ...