Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001761
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => [1] => 0
[1,1] => [1,1,0,0]
 => [2,1] => [2,1] => 1
[1,2] => [1,0,1,0]
 => [1,2] => [1,2] => 0
[2,1] => [1,0,1,0]
 => [1,2] => [1,2] => 0
[1,1,1] => [1,1,1,0,0,0]
 => [3,1,2] => [3,1,2] => 1
[1,1,2] => [1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => 2
[1,2,1] => [1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => 2
[2,1,1] => [1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => 2
[1,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[1,3,1] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[3,1,1] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1
[2,1,2] => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1
[2,2,1] => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [4,1,3,2] => 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => 3
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => 3
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => 3
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => 3
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4,2,1,3] => 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 3
Description
The maximal multiplicity of a letter in a reduced word of a permutation. For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Matching statistic: St000317
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 43%
Values
[1] => [1,0]
 => [(1,2)]
 => [2,1] => 0
[1,1] => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1
[1,2] => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0
[2,1] => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0
[1,1,1] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 1
[1,1,2] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2
[1,2,1] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2
[2,1,1] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2
[1,1,3] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => 1
[1,3,1] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => 1
[3,1,1] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => 1
[1,2,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1
[2,1,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1
[2,2,1] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[1,3,2] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[2,1,3] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[2,3,1] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[3,1,2] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[3,2,1] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? = 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 2
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 2
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 2
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 2
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 3
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 3
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 3
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 3
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? = 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? = 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? = 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? = 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[3,2,1,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => ? = 1
Description
The cycle descent number of a permutation. Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.