Riffle shuffle permutation

In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is a permutation of a set of ${\displaystyle n}$ ordered items that can be obtained by a single riffle shuffle, in which a sorted deck of ${\displaystyle n}$ cards (increasing top-to-bottom) is cut into two packets and then the two packets are interleaved (e.g. by moving cards one at a time from the bottom of one or the other of the packets to the top of the sorted deck). As a special case of this, a ${\displaystyle (p,q)}$-shuffle, for numbers ${\displaystyle p}$ and ${\displaystyle q}$ with ${\displaystyle p+q=n}$, is a riffle in which the first packet has ${\displaystyle p}$ cards and the second packet has ${\displaystyle q}$ cards.

Considering a permutation as a bijective function ${\displaystyle \pi }$ from the set ${\displaystyle \{1,2,\ldots ,n\}}$ to itself, a riffle shuffle is defined as containing only 1 or 2 maximal rising sequences, meaning ${\displaystyle \{1,2,\ldots ,n\}}$ can be decomposed into two disjoint subsets ${\displaystyle \{i_{1}<\cdots <i_{p}\}}$ and ${\displaystyle \{j_{1}<\cdots <j_{q}\}}$ with

${\displaystyle \pi (i_{1})<\pi (i_{2})<\cdots <\pi (i_{p})}$ and ${\displaystyle \pi (j_{1})<\pi (j_{2})<\cdots <\pi (j_{q})}$.

A permutation with only 1 maximal rising sequence is the identity permutation.

The inverse permutation ${\displaystyle \tau =\pi ^{-1}}$of a riffle shuffle is known as Grassmannian permutation, defined by

${\displaystyle \tau (1)<\ldots <\tau (p)\ \ \ {\text{and}}\ \ \ \tau (p+1)<\ldots <\tau (p+q),}$

having one descent ${\displaystyle \tau (p)>\tau (p+1)}$, or zero descents if ${\displaystyle \tau }$ is the identity. In Schubert calculus, these index Schubert varieties in a Grassmannian space.

A permutation ${\displaystyle \pi }$ which is both a riffle shuffle and Grassmannian (i.e. both ${\displaystyle \pi }$ and its inverse are Grassmannian, or equivalently both are riffle shuffles), is called bigrassmannian or an invertible shuffle.