  We investigate the 1D version of the notable Bressan's mixing conjecture, and
introduce various formulation in the classical optimal transport setting, the
branched optimal transport setting and a combinatorial optimization. In the
discrete case of the combinatorial problem, we prove the number of admissible
solutions is on the Catalan number. Our investigation sheds light on the
intricate relationship between mixing problem in the fluid dynamics and many
other popular fields, leaving many interesting open questions in both
theoretical and practical applications across disciplines.
