Constructions of Totally Non-Negative Pfaffian

The totally non-negative pfaffian (TNNP) is define for a skew-symmetric matrix such that all the sub-pfaffians are non-negative. It appears in the pfaffian structure of $\tau$-function for the non-singular web solitons of the BKP equation . One constructs TNNP using the Perfect matching, chord diagram and the Dyck paths enumeration.Its tridiagonal form is also investigated.