Your data matches 44 different statistics following compositions of up to 3 maps.
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Mp00010: Binary trees to ordered tree: left child = left brotherOrdered trees
St000522: Ordered trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [[]]
=> 1
[.,[.,.]]
=> [[[]]]
=> 1
[[.,.],.]
=> [[],[]]
=> 1
[.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[.,[[.,.],.]]
=> [[[],[]]]
=> 1
[[.,.],[.,.]]
=> [[],[[]]]
=> 2
[[.,[.,.]],.]
=> [[[]],[]]
=> 2
[[[.,.],.],.]
=> [[],[],[]]
=> 1
[.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 1
[.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2
[.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 1
[[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> 2
[[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2
[[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 2
[[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 2
[[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> 2
[[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 2
[[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 2
[[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> 2
[.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [[],[[[],[]]]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [[],[[],[[]]]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [[],[[[]],[]]]
=> 3
[[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [[[]],[[],[]]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 2
[[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [[[[]]],[[]]]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [[[],[]],[[]]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 3
[[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> 3
[[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 2
Description
The number of 1-protected nodes of a rooted tree. This is the number of nodes with minimal distance one to a leaf.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00326: Permutations weak order rowmotionPermutations
Mp00088: Permutations Kreweras complementPermutations
St000092: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [1,2] => [2,1] => 1
[[.,.],.]
=> [1,2] => [2,1] => [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [2,3,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => [3,2,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => [2,1,3] => 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [3,1,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,3,2] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1,2,4] => [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,1,3] => [4,2,1,3] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,1,4] => [4,2,3,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,1,4] => [4,3,2,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,3,4,2] => [2,1,3,4] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,2,4] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,2,4,3] => [2,3,1,4] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [3,4,1,2] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,2,1,3] => [4,3,1,2] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,1,3,2] => [3,1,4,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [4,1,3,2] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1,2,3,5] => [3,4,5,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,5,1,2,4] => [4,5,2,1,3] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,4,1,2,5] => [4,5,2,3,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,3,1,2,5] => [4,5,3,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,4,5,1,3] => [5,2,1,3,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,5,1,3] => [5,3,1,2,4] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,3,5,1,4] => [5,2,3,1,4] => 3
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,5,3,1,4] => [5,2,4,1,3] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,3,4,1,5] => [5,2,3,4,1] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2,4,1,5] => [5,3,2,4,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,1,5] => [5,2,4,3,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,2,3,1,5] => [5,3,4,2,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,3,2,1,5] => [5,4,3,2,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,3,4,5,2] => [2,1,3,4,5] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,1,3,5,2] => [3,1,4,2,5] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,5,1,4,2] => [4,1,2,5,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,4,1,5,2] => [4,1,2,3,5] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,3,1,5,2] => [4,1,3,2,5] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,2,4,5,3] => [2,3,1,4,5] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,1,2,5,3] => [3,4,1,2,5] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,4,5,3,2] => [2,1,5,3,4] => 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,5,3,2] => [3,1,5,2,4] => 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,5,4,1,3] => [5,2,1,4,3] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,3,5,4,2] => [2,1,3,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [2,1,5,4,3] => 2
Description
The number of outer peaks of a permutation. An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$. In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000093
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St000093: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[.,.],.],.]
=> [1,2,3] => ([],3)
=> ([],1)
=> 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)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000099
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000099: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [2,1] => [1,2] => 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 2
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,3,2] => 2
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [1,2,3] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,2,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,4,2,3] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,4,2,3] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,3,1] => [1,4,2,3] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,4,3,2] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,4,2,1] => [1,3,2,4] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => [1,2,4,3] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,4,1] => [1,3,4,2] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,3,1,4] => [1,2,3,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,5,2,4,3] => 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,5,2,4,3] => 3
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,2,3,1] => [1,5,2,4,3] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,5,3,2,4] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,5,3,2,4] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,3,4,2,1] => [1,5,2,3,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,4,1,3] => [1,5,3,4,2] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,4,1] => [1,5,2,3,4] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,5,2,3,4] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,5,4,2,3] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,3,1,4] => [1,5,4,2,3] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,5,4,3,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [4,5,3,2,1] => [1,4,2,5,3] => 3
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,5,4,1,2] => [1,3,4,2,5] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,5,2,4,1] => [1,3,2,5,4] => 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,5,2,1,4] => [1,3,2,5,4] => 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,1,3,4] => [1,2,5,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [4,3,5,2,1] => [1,4,2,3,5] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2,5,1,4] => [1,3,5,4,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,3,2,5,1] => [1,4,5,2,3] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,2,3,5,1] => [1,4,5,2,3] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,4,2,5,1] => [1,3,2,4,5] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [3,2,4,5,1] => [1,3,4,5,2] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 1
Description
The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000786
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St000786: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[.,.],.],.]
=> [1,2,3] => ([],3)
=> ([],1)
=> 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)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
Description
The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St001337
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St001337: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[.,.],.],.]
=> [1,2,3] => ([],3)
=> ([],1)
=> 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)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
Description
The upper domination number of a graph. This is the maximum cardinality of a minimal dominating set of $G$. The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St001338: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[[[.,.],.],.]
=> [1,2,3] => ([],3)
=> ([],1)
=> 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)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
Description
The upper irredundance number of a graph. A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$. The upper irredundance number is the largest size of a maximal irredundant set. The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St000023
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000023: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,4,2,3] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,4,3,2] => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => [1,2,4,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,4,1] => [1,3,4,2] => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,5,2,4,3] => 2 = 3 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,5,2,4,3] => 2 = 3 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,2,3,1] => [1,5,2,4,3] => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,5,3,2,4] => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,5,3,2,4] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,3,4,2,1] => [1,5,2,3,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,4,1,3] => [1,5,3,4,2] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,4,1] => [1,5,2,3,4] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,5,2,3,4] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,5,4,2,3] => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,5,4,2,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,3,1,4] => [1,5,4,2,3] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,5,4,3,2] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [4,5,3,2,1] => [1,4,2,5,3] => 2 = 3 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,5,4,1,2] => [1,3,4,2,5] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,5,2,4,1] => [1,3,2,5,4] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,5,2,1,4] => [1,3,2,5,4] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,1,3,4] => [1,2,5,4,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [4,3,5,2,1] => [1,4,2,3,5] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2,5,1,4] => [1,3,5,4,2] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,3,2,5,1] => [1,4,5,2,3] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,2,3,5,1] => [1,4,5,2,3] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,4,2,5,1] => [1,3,2,4,5] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [3,2,4,5,1] => [1,3,4,5,2] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
Description
The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00086: Permutations first fundamental transformationPermutations
Mp00252: Permutations restrictionPermutations
St000035: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [2,1] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,1] => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,2] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [1,3,2] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [2,1,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,2,1] => 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,1] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 2 = 3 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
Description
The number of left outer peaks of a permutation. A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$. In other words, it is a peak in the word $[0,w_1,..., w_n]$. This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00252: Permutations restrictionPermutations
Mp00069: Permutations complementPermutations
St000337: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [] => [] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [1] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 1 = 2 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 2 = 3 - 1
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001928The number of non-overlapping descents in a permutation. St000568The hook number of a binary tree. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St000480The number of lower covers of a partition in dominance order. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000660The number of rises of length at least 3 of a Dyck path. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001470The cyclic holeyness of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001597The Frobenius rank of a skew partition. St000264The girth of a graph, which is not a tree. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000455The second largest eigenvalue of a graph if it is integral. St000628The balance of a binary word. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001060The distinguishing index of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St000758The length of the longest staircase fitting into an integer composition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001423The number of distinct cubes in a binary word. St001823The Stasinski-Voll length of a signed permutation. St000920The logarithmic height of a Dyck path.