- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions

Images

=>
            
              Cc0005;cc-rep-1Cc0009;cc-rep-3
            

            [1]=>[1,0]=>[2,1]=>{{1,2}}
[1,1]=>[1,0,1,0]=>[3,1,2]=>{{1,2,3}}
[2]=>[1,1,0,0]=>[2,3,1]=>{{1,2,3}}
[1,1,1]=>[1,0,1,0,1,0]=>[4,1,2,3]=>{{1,2,3,4}}
[1,2]=>[1,0,1,1,0,0]=>[3,1,4,2]=>{{1,2,3,4}}
[2,1]=>[1,1,0,0,1,0]=>[2,4,1,3]=>{{1,2,3,4}}
[3]=>[1,1,1,0,0,0]=>[2,3,4,1]=>{{1,2,3,4}}
[1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[5,1,2,3,4]=>{{1,2,3,4,5}}
[1,1,2]=>[1,0,1,0,1,1,0,0]=>[4,1,2,5,3]=>{{1,2,3,4,5}}
[1,2,1]=>[1,0,1,1,0,0,1,0]=>[3,1,5,2,4]=>{{1,2,3,4,5}}
[1,3]=>[1,0,1,1,1,0,0,0]=>[3,1,4,5,2]=>{{1,2,3,4,5}}
[2,1,1]=>[1,1,0,0,1,0,1,0]=>[2,5,1,3,4]=>{{1,2,3,4,5}}
[2,2]=>[1,1,0,0,1,1,0,0]=>[2,4,1,5,3]=>{{1,2,3,4,5}}
[3,1]=>[1,1,1,0,0,0,1,0]=>[2,3,5,1,4]=>{{1,2,3,4,5}}
[4]=>[1,1,1,1,0,0,0,0]=>[2,3,4,5,1]=>{{1,2,3,4,5}}
[1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[6,1,2,3,4,5]=>{{1,2,3,4,5,6}}
[1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[5,1,2,3,6,4]=>{{1,2,3,4,5,6}}
[1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[4,1,2,6,3,5]=>{{1,2,3,4,5,6}}
[1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[4,1,2,5,6,3]=>{{1,2,3,4,5,6}}
[1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[3,1,6,2,4,5]=>{{1,2,3,4,5,6}}
[1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[3,1,5,2,6,4]=>{{1,2,3,4,5,6}}
[1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[3,1,4,6,2,5]=>{{1,2,3,4,5,6}}
[1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[3,1,4,5,6,2]=>{{1,2,3,4,5,6}}
[2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[2,6,1,3,4,5]=>{{1,2,3,4,5,6}}
[2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[2,5,1,3,6,4]=>{{1,2,3,4,5,6}}
[2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[2,4,1,6,3,5]=>{{1,2,3,4,5,6}}
[2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[2,4,1,5,6,3]=>{{1,2,3,4,5,6}}
[3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[2,3,6,1,4,5]=>{{1,2,3,4,5,6}}
[3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[2,3,5,1,6,4]=>{{1,2,3,4,5,6}}
[4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[2,3,4,6,1,5]=>{{1,2,3,4,5,6}}
[5]=>[1,1,1,1,1,0,0,0,0,0]=>[2,3,4,5,6,1]=>{{1,2,3,4,5,6}}
[1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[7,1,2,3,4,5,6]=>{{1,2,3,4,5,6,7}}
[1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[6,1,2,3,4,7,5]=>{{1,2,3,4,5,6,7}}
[1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[5,1,2,3,7,4,6]=>{{1,2,3,4,5,6,7}}
[1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[5,1,2,3,6,7,4]=>{{1,2,3,4,5,6,7}}
[1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[4,1,2,7,3,5,6]=>{{1,2,3,4,5,6,7}}
[1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[4,1,2,6,3,7,5]=>{{1,2,3,4,5,6,7}}
[1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[4,1,2,5,7,3,6]=>{{1,2,3,4,5,6,7}}
[1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[4,1,2,5,6,7,3]=>{{1,2,3,4,5,6,7}}
[1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[3,1,7,2,4,5,6]=>{{1,2,3,4,5,6,7}}
[1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[3,1,6,2,4,7,5]=>{{1,2,3,4,5,6,7}}
[1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[3,1,5,2,7,4,6]=>{{1,2,3,4,5,6,7}}
[1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[3,1,5,2,6,7,4]=>{{1,2,3,4,5,6,7}}
[1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[3,1,4,7,2,5,6]=>{{1,2,3,4,5,6,7}}
[1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[3,1,4,6,2,7,5]=>{{1,2,3,4,5,6,7}}
[1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[3,1,4,5,7,2,6]=>{{1,2,3,4,5,6,7}}
[1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[3,1,4,5,6,7,2]=>{{1,2,3,4,5,6,7}}
[2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,7,1,3,4,5,6]=>{{1,2,3,4,5,6,7}}
[2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,6,1,3,4,7,5]=>{{1,2,3,4,5,6,7}}
[2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,5,1,3,7,4,6]=>{{1,2,3,4,5,6,7}}
[2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,5,1,3,6,7,4]=>{{1,2,3,4,5,6,7}}
[2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,4,1,7,3,5,6]=>{{1,2,3,4,5,6,7}}
[2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,4,1,6,3,7,5]=>{{1,2,3,4,5,6,7}}
[2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,4,1,5,7,3,6]=>{{1,2,3,4,5,6,7}}
[2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,4,1,5,6,7,3]=>{{1,2,3,4,5,6,7}}
[3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[2,3,7,1,4,5,6]=>{{1,2,3,4,5,6,7}}
[3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[2,3,6,1,4,7,5]=>{{1,2,3,4,5,6,7}}
[3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[2,3,5,1,7,4,6]=>{{1,2,3,4,5,6,7}}
[3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[2,3,5,1,6,7,4]=>{{1,2,3,4,5,6,7}}
[4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[2,3,4,7,1,5,6]=>{{1,2,3,4,5,6,7}}
[4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[2,3,4,6,1,7,5]=>{{1,2,3,4,5,6,7}}
[5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[2,3,4,5,7,1,6]=>{{1,2,3,4,5,6,7}}
[6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[2,3,4,5,6,7,1]=>{{1,2,3,4,5,6,7}}
          

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

to cycle type

Description

Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].