searching the database
Your data matches 60 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000823
St000823: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 2
{{1},{2}}
=> 1
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 1
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 3
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 2
{{1,3,4},{2}}
=> 3
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 1
{{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 4
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 3
{{1,2,4,5},{3}}
=> 4
{{1,2,4},{3,5}}
=> 3
{{1,2,4},{3},{5}}
=> 2
{{1,2,5},{3,4}}
=> 3
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 3
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 2
{{1,3,4,5},{2}}
=> 4
{{1,3,4},{2,5}}
=> 3
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 3
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 2
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 1
Description
The number of unsplittable factors of the set partition.
Let $\pi$ be a set partition of $m$ into $k$ parts and $\sigma$ a set partition of $n$ into $\ell$ parts. Then their split product is the set partition of $m+n$ with blocks
$$ B_1 \cup (C_1+m),\dots,B_k \cup (C_k+m), C_{k+1}+m,\dots,C_\ell+m $$
if $k\leq \ell$ and
$$ B_1 \cup (C_1+m),\dots,B_\ell \cup (C_\ell+m), B_{\ell+1},\dots,B_k $$
otherwise.
Matching statistic: St000011
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000288
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => 0 => 0 = 1 - 1
{{1},{2}}
=> [1,2] => 1 => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => 00 => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => 01 => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => 00 => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => 10 => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => 11 => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => 000 => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => 001 => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => 000 => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => 010 => 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 011 => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => 000 => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => 000 => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 001 => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => 000 => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => 100 => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => 101 => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 000 => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => 100 => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => 110 => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 111 => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 0000 => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0001 => 1 = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0000 => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0010 => 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0011 => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 0000 => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0000 => 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 0001 => 1 = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 0000 => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0100 => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0101 => 2 = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 0000 => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 0100 => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0110 => 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0111 => 3 = 4 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 0000 => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 0000 => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 0001 => 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 0000 => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0000 => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 0001 => 1 = 2 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 0000 => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 0000 => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 0010 => 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0011 => 2 = 3 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 0000 => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 0000 => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 0001 => 1 = 2 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000010
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ 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] => ([(0,1)],2)
=> [2]
=> 1
{{1},{2}}
=> [1,2] => ([],2)
=> [1,1]
=> 2
{{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
{{1,2},{3}}
=> [2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
{{1},{2,3}}
=> [1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2
{{1},{2},{3}}
=> [1,2,3] => ([],3)
=> [1,1,1]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
Description
The length of the partition.
Matching statistic: St000025
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000383
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => ([(0,1)],2)
=> [1,1] => 1
{{1},{2}}
=> [1,2] => ([],2)
=> [2] => 2
{{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1
{{1,2},{3}}
=> [2,1,3] => ([(1,2)],3)
=> [1,2] => 2
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 1
{{1},{2,3}}
=> [1,3,2] => ([(1,2)],3)
=> [1,2] => 2
{{1},{2},{3}}
=> [1,2,3] => ([],3)
=> [3] => 3
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(2,3)],4)
=> [1,3] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(2,3)],4)
=> [1,3] => 3
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> [1,3] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([],4)
=> [4] => 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,3] => 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,3] => 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(3,4)],5)
=> [1,4] => 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 2
Description
The last part of an integer composition.
Matching statistic: St000675
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001462
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [[1],[2]]
=> 1
{{1},{2}}
=> [1,2] => [1,2] => [[1,2]]
=> 2
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 2
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St000234
(load all 133 compositions to match this statistic)
(load all 133 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => [4,3,2,1] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,3,1] => [4,3,2,1] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 1 = 2 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 3 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 3 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1 = 2 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2 = 3 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000439The position of the first down step of a Dyck path. St001479The number of bridges of a graph. St000916The packing number of a graph. St000717The number of ordinal summands of a poset. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000553The number of blocks of a graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000237The number of small exceedances. St000260The radius of a connected graph. St001461The number of topologically connected components of the chord diagram of a permutation. St000843The decomposition number of a perfect matching. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000259The diameter of a connected graph. St000056The decomposition (or block) number of a permutation. St000287The number of connected components of a graph. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000181The number of connected components of the Hasse diagram for the poset. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001889The size of the connectivity set of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001904The length of the initial strictly increasing segment of a parking function. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001937The size of the center of a parking function.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!