Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000844
(load all 246 compositions to match this statistic)
Mapping
St000844: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 3
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 3
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 4
[2,4,1,3] => 4
[2,4,3,1] => 4
[3,1,2,4] => 3
[3,1,4,2] => 4
[3,2,1,4] => 3
[3,2,4,1] => 4
[3,4,1,2] => 4
[3,4,2,1] => 4
[4,1,2,3] => 4
[4,1,3,2] => 4
[4,2,1,3] => 4
[4,2,3,1] => 4
[4,3,1,2] => 4
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 3
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 3
[1,3,4,5,2] => 4
[1,3,5,2,4] => 4
[1,3,5,4,2] => 4
[1,4,2,3,5] => 3
[1,4,2,5,3] => 4
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 4
[1,4,5,3,2] => 4
Description
The size of the largest block in the direct sum decomposition of a permutation. A component of a permutation $\pi$ is a set of consecutive numbers $\{a,a+1,\dots, b\}$ such that $a\leq \pi(i) \leq b$ for all $a\leq i\leq b$. This statistic is the size of the largest component which does not properly contain another component.
Matching statistic: St001268
(load all 22 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St001268: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => ([(0,1)],2)
 => 1
[2,1] => ([],2)
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => ([(1,2)],3)
 => 3
[3,1,2] => ([(1,2)],3)
 => 3
[3,2,1] => ([],3)
 => 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 4
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 4
[2,4,3,1] => ([(1,2),(1,3)],4)
 => 4
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 4
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 4
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[3,4,2,1] => ([(2,3)],4)
 => 4
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 4
[4,1,3,2] => ([(1,2),(1,3)],4)
 => 4
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 4
[4,2,3,1] => ([(2,3)],4)
 => 4
[4,3,1,2] => ([(2,3)],4)
 => 4
[4,3,2,1] => ([],4)
 => 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 3
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 3
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 3
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 4
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 4
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 3
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 4
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 4
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 4
Description
The size of the largest ordinal summand in the poset. The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$. This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.
Matching statistic: St000147
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => ([],2)
 => [1,1]
 => 1
[2,1] => ([(0,1)],2)
 => [2]
 => 2
[1,2,3] => ([],3)
 => [1,1,1]
 => 1
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => 2
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => 1
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
Description
The largest part of an integer partition.
Matching statistic: St000381
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
 => [1,1] => 1
[2,1] => [1,1,0,0]
 => [2] => 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,2] => 2
[2,1,3] => [1,1,0,0,1,0]
 => [2,1] => 2
[2,3,1] => [1,1,0,1,0,0]
 => [3] => 3
[3,1,2] => [1,1,1,0,0,0]
 => [3] => 3
[3,2,1] => [1,1,1,0,0,0]
 => [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [4] => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,1] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [4] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,1] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [4] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 4
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 4
Description
The largest part of an integer composition.
Matching statistic: St000209
(load all 42 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[2,1] => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3 = 4 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3 = 4 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3 = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3 = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2 = 3 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2 = 3 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 3 = 4 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 3 = 4 - 1
Description
Maximum difference of elements in cycles. Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$. The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St000392
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => 0 => 0 = 1 - 1
[2,1] => 0 => 1 => 1 = 2 - 1
[1,2,3] => 11 => 00 => 0 = 1 - 1
[1,3,2] => 10 => 01 => 1 = 2 - 1
[2,1,3] => 01 => 10 => 1 = 2 - 1
[2,3,1] => 00 => 11 => 2 = 3 - 1
[3,1,2] => 00 => 11 => 2 = 3 - 1
[3,2,1] => 00 => 11 => 2 = 3 - 1
[1,2,3,4] => 111 => 000 => 0 = 1 - 1
[1,2,4,3] => 110 => 001 => 1 = 2 - 1
[1,3,2,4] => 101 => 010 => 1 = 2 - 1
[1,3,4,2] => 100 => 011 => 2 = 3 - 1
[1,4,2,3] => 100 => 011 => 2 = 3 - 1
[1,4,3,2] => 100 => 011 => 2 = 3 - 1
[2,1,3,4] => 011 => 100 => 1 = 2 - 1
[2,1,4,3] => 010 => 101 => 1 = 2 - 1
[2,3,1,4] => 001 => 110 => 2 = 3 - 1
[2,3,4,1] => 000 => 111 => 3 = 4 - 1
[2,4,1,3] => 000 => 111 => 3 = 4 - 1
[2,4,3,1] => 000 => 111 => 3 = 4 - 1
[3,1,2,4] => 001 => 110 => 2 = 3 - 1
[3,1,4,2] => 000 => 111 => 3 = 4 - 1
[3,2,1,4] => 001 => 110 => 2 = 3 - 1
[3,2,4,1] => 000 => 111 => 3 = 4 - 1
[3,4,1,2] => 000 => 111 => 3 = 4 - 1
[3,4,2,1] => 000 => 111 => 3 = 4 - 1
[4,1,2,3] => 000 => 111 => 3 = 4 - 1
[4,1,3,2] => 000 => 111 => 3 = 4 - 1
[4,2,1,3] => 000 => 111 => 3 = 4 - 1
[4,2,3,1] => 000 => 111 => 3 = 4 - 1
[4,3,1,2] => 000 => 111 => 3 = 4 - 1
[4,3,2,1] => 000 => 111 => 3 = 4 - 1
[1,2,3,4,5] => 1111 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => 1110 => 0001 => 1 = 2 - 1
[1,2,4,3,5] => 1101 => 0010 => 1 = 2 - 1
[1,2,4,5,3] => 1100 => 0011 => 2 = 3 - 1
[1,2,5,3,4] => 1100 => 0011 => 2 = 3 - 1
[1,2,5,4,3] => 1100 => 0011 => 2 = 3 - 1
[1,3,2,4,5] => 1011 => 0100 => 1 = 2 - 1
[1,3,2,5,4] => 1010 => 0101 => 1 = 2 - 1
[1,3,4,2,5] => 1001 => 0110 => 2 = 3 - 1
[1,3,4,5,2] => 1000 => 0111 => 3 = 4 - 1
[1,3,5,2,4] => 1000 => 0111 => 3 = 4 - 1
[1,3,5,4,2] => 1000 => 0111 => 3 = 4 - 1
[1,4,2,3,5] => 1001 => 0110 => 2 = 3 - 1
[1,4,2,5,3] => 1000 => 0111 => 3 = 4 - 1
[1,4,3,2,5] => 1001 => 0110 => 2 = 3 - 1
[1,4,3,5,2] => 1000 => 0111 => 3 = 4 - 1
[1,4,5,2,3] => 1000 => 0111 => 3 = 4 - 1
[1,4,5,3,2] => 1000 => 0111 => 3 = 4 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000503
(load all 9 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
 => {{1},{2}}
 => 0 = 1 - 1
[2,1] => [1,1,0,0]
 => {{1,2}}
 => 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 3 = 4 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 3 = 4 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 3 = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 3 = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 3 = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 3 = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 2 = 3 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 2 = 3 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 3 = 4 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 3 = 4 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 3 = 4 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 3 = 4 - 1
Description
The maximal difference between two elements in a common block.
Matching statistic: St000956
(load all 33 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
 => [1,2] => 0 = 1 - 1
[2,1] => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3 = 4 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3 = 4 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3 = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3 = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2 = 3 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2 = 3 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 3 = 4 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 3 = 4 - 1
Description
The maximal displacement of a permutation. This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$. This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St001120
(load all 3 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St001120: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => ([],2)
 => ([],2)
 => 0 = 1 - 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,2,3] => ([],3)
 => ([],3)
 => 0 = 1 - 1
[1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
[2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 2 - 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,2,3,4] => ([],4)
 => ([],4)
 => 0 = 1 - 1
[1,2,4,3] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[1,3,2,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,1,3,4] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 1 = 2 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0 = 1 - 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1 = 2 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1 = 2 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
Description
The length of a longest path in a graph.
Matching statistic: St000010
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => ([],2)
 => [1,1]
 => [2]
 => 1
[2,1] => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[1,2,3] => ([],3)
 => [1,1,1]
 => [3]
 => 1
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => [2,1]
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 3
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 3
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => [4]
 => 1
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 4
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => [5]
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
Description
The length of the partition.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000141The maximum drop size of a permutation. St000171The degree of the graph. St000442The maximal area to the right of an up step of a Dyck path. St000662The staircase size of the code of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001330The hat guessing number of a graph. St001644The dimension of a graph. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001877Number of indecomposable injective modules with projective dimension 2. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset.