Your data matches 136 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00223: Permutations runsortPermutations
Mp00109: Permutations descent wordBinary words
St000392: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 1
[2,1,3] => [1,3,2] => 01 => 1
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,2,3] => 00 => 0
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,3,4,2] => 001 => 1
[1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,4,2,3] => 010 => 1
[2,1,3,4] => [1,3,4,2] => 001 => 1
[2,1,4,3] => [1,4,2,3] => 010 => 1
[2,3,1,4] => [1,4,2,3] => 010 => 1
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,3,2,4] => 010 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,2,4,3] => 001 => 1
[3,1,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,4,2,3] => 010 => 1
[3,2,4,1] => [1,2,4,3] => 001 => 1
[3,4,1,2] => [1,2,3,4] => 000 => 0
[3,4,2,1] => [1,2,3,4] => 000 => 0
[4,1,2,3] => [1,2,3,4] => 000 => 0
[4,1,3,2] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [1,3,2,4] => 010 => 1
[4,2,3,1] => [1,2,3,4] => 000 => 0
[4,3,1,2] => [1,2,3,4] => 000 => 0
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => [1,2,5,3,4] => 0010 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => [1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => [1,3,5,2,4] => 0010 => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => [1,4,2,5,3] => 0101 => 1
[1,4,3,2,5] => [1,4,2,5,3] => 0101 => 1
[1,4,3,5,2] => [1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => [1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => [1,4,5,2,3] => 0010 => 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000306
Mp00223: Permutations runsortPermutations
Mp00204: Permutations LLPSInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000306: Dyck paths ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,5,4,7,6,3,8,1] => [1,2,5,3,8,4,7,6] => ?
=> ?
=> ? = 1 + 1
[2,4,3,7,6,5,8,1] => [1,2,4,3,7,5,8,6] => ?
=> ?
=> ? = 1 + 1
[2,1,4,5,8,7,6,3] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[2,1,3,4,6,7,8,5] => [1,3,4,6,7,8,2,5] => ?
=> ?
=> ? = 1 + 1
[3,2,1,4,5,8,7,6] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[3,2,4,1,6,7,5,8] => [1,6,7,2,4,3,5,8] => ?
=> ?
=> ? = 1 + 1
[2,1,4,5,8,6,7,3] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[2,1,4,5,8,3,6,7] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[2,5,1,3,7,8,4,6] => [1,3,7,8,2,5,4,6] => ?
=> ?
=> ? = 1 + 1
[2,3,6,1,4,5,8,7] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[3,2,6,1,4,5,7,8] => [1,4,5,7,8,2,6,3] => ?
=> ?
=> ? = 1 + 1
[7,3,4,8,1,2,5,6] => [1,2,5,6,3,4,8,7] => ?
=> ?
=> ? = 1 + 1
[8,3,5,1,6,7,2,4] => [1,6,7,2,4,3,5,8] => ?
=> ?
=> ? = 1 + 1
[4,8,1,6,7,3,5,2] => [1,6,7,2,3,5,4,8] => ?
=> ?
=> ? = 1 + 1
[7,6,1,4,5,8,3,2] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[1,4,5,8,7,3,6,2] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[2,1,6,7,4,8,5,3] => [1,6,7,2,3,4,8,5] => ?
=> ?
=> ? = 1 + 1
[2,3,8,1,4,6,7,5] => [1,4,6,7,2,3,8,5] => ?
=> ?
=> ? = 1 + 1
[4,8,2,3,5,1,6,7] => [1,6,7,2,3,5,4,8] => ?
=> ?
=> ? = 1 + 1
[6,5,8,2,4,3,7,1] => [1,2,4,3,7,5,8,6] => ?
=> ?
=> ? = 1 + 1
[4,7,6,3,8,1,2,5] => [1,2,5,3,8,4,7,6] => ?
=> ?
=> ? = 1 + 1
[6,3,7,5,8,1,2,4] => [1,2,4,3,7,5,8,6] => ?
=> ?
=> ? = 1 + 1
[8,4,1,6,7,2,3,5] => [1,6,7,2,3,5,4,8] => ?
=> ?
=> ? = 1 + 1
[6,7,1,4,5,8,2,3] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[7,4,8,3,2,5,6,1] => [1,2,5,6,3,4,8,7] => ?
=> ?
=> ? = 1 + 1
[7,4,8,2,5,6,3,1] => [1,2,5,6,3,4,8,7] => ?
=> ?
=> ? = 1 + 1
[3,7,5,8,6,2,4,1] => [1,2,4,3,7,5,8,6] => ?
=> ?
=> ? = 1 + 1
[9,7,10,5,6,8,1,2,3,4] => [1,2,3,4,5,6,8,7,10,9] => ?
=> ?
=> ? = 1 + 1
[6,3,2,1,4,5,8,7] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[1,6,8,3,4,7,5,2] => [1,6,8,2,3,4,7,5] => ?
=> ?
=> ? = 1 + 1
[1,6,7,5,8,2,4,3] => [1,6,7,2,4,3,5,8] => ?
=> ?
=> ? = 1 + 1
[3,1,4,5,8,6,7,2] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[2,1,6,8,3,4,7,5] => [1,6,8,2,3,4,7,5] => ?
=> ?
=> ? = 1 + 1
[8,5,2,4,1,6,7,3] => [1,6,7,2,4,3,5,8] => ?
=> ?
=> ? = 1 + 1
[8,2,3,5,1,6,7,4] => [1,6,7,2,3,5,4,8] => ?
=> ?
=> ? = 1 + 1
[1,2,5,4,7,3,8,6] => [1,2,5,3,8,4,7,6] => ?
=> ?
=> ? = 1 + 1
[4,3,7,1,8,2,5,6] => [1,8,2,5,6,3,7,4] => ?
=> ?
=> ? = 1 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [1,3,4,5,6,7,8,9,10,11,2] => [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[8,3,6,1,2,5,7,4] => [1,2,5,7,3,6,4,8] => ?
=> ?
=> ? = 1 + 1
[7,2,3,1,4,5,8,6] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1 + 1
[11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[10,1,2,3,4,5,6,7,8,11,9] => [1,2,3,4,5,6,7,8,11,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[11,10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[6,2,5,1,3,7,8,4] => [1,3,7,8,2,5,4,6] => ?
=> ?
=> ? = 1 + 1
[4,2,5,6,1,8,3,7] => [1,8,2,5,6,3,7,4] => ?
=> ?
=> ? = 1 + 1
[3,2,1,6,8,5,4,7] => [1,6,8,2,3,4,7,5] => ?
=> ?
=> ? = 1 + 1
[10,9,8,7,6,5,4,3,2,1,11] => [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,11,10,9,8,7,6,5,4,3,2] => [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[2,3,4,5,6,7,8,9,10,11,1] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Mp00223: Permutations runsortPermutations
Mp00064: Permutations reversePermutations
Mp00071: Permutations descent compositionInteger compositions
St000381: Integer compositions ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,3,1] => [2,1] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,3,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 2 = 1 + 1
[8,7,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[7,8,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[7,8,2,3,4,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[8,6,5,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[8,5,6,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[7,4,3,2,1,5,6,8] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[1,2,4,7,8,6,5,3] => [1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ? = 1 + 1
[3,2,1,5,6,8,7,4] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[2,4,3,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[3,2,4,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[3,2,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[2,3,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[3,2,4,1,5,8,7,6] => [1,5,8,2,4,3,6,7] => [7,6,3,4,2,8,5,1] => ? => ? = 1 + 1
[4,3,2,1,5,6,8,7] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[2,3,4,1,5,6,8,7] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[3,2,4,1,7,6,5,8] => [1,7,2,4,3,5,8,6] => [6,8,5,3,4,2,7,1] => ? => ? = 1 + 1
[3,2,4,1,5,7,6,8] => [1,5,7,2,4,3,6,8] => [8,6,3,4,2,7,5,1] => ? => ? = 1 + 1
[3,2,5,4,1,6,7,8] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[1,3,8,5,4,6,7,2] => [1,3,8,2,4,6,7,5] => [5,7,6,4,2,8,3,1] => ? => ? = 1 + 1
[1,7,5,4,3,6,8,2] => [1,7,2,3,6,8,4,5] => [5,4,8,6,3,2,7,1] => ? => ? = 1 + 1
[5,2,4,3,1,6,7,8] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[2,4,1,6,7,8,3,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[2,5,1,6,7,8,3,4] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[2,1,6,3,7,4,5,8] => [1,6,2,3,7,4,5,8] => [8,5,4,7,3,2,6,1] => ? => ? = 1 + 1
[2,3,6,1,4,7,5,8] => [1,4,7,2,3,6,5,8] => [8,5,6,3,2,7,4,1] => ? => ? = 1 + 1
[3,1,4,6,8,2,5,7] => [1,4,6,8,2,5,7,3] => [3,7,5,2,8,6,4,1] => ? => ? = 1 + 1
[4,6,1,2,7,8,3,5] => [1,2,7,8,3,5,4,6] => [6,4,5,3,8,7,2,1] => ? => ? = 1 + 1
[4,1,2,6,7,3,8,5] => [1,2,6,7,3,8,4,5] => [5,4,8,3,7,6,2,1] => ? => ? = 1 + 1
[1,2,4,5,7,3,8,6] => [1,2,4,5,7,3,8,6] => [6,8,3,7,5,4,2,1] => ? => ? = 1 + 1
[2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,2,4,1] => ? => ? = 1 + 1
[8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[9,7,5,10,3,8,2,6,1,4] => [1,4,2,6,3,8,5,10,7,9] => [9,7,10,5,8,3,6,2,4,1] => [1,2,2,2,2,1] => ? = 1 + 1
[4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[6,3,5,1,2,4,7,8] => [1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ? = 1 + 1
[8,3,5,1,2,4,6,7] => [1,2,4,6,7,3,5,8] => [8,5,3,7,6,4,2,1] => ? => ? = 1 + 1
[8,1,4,7,6,2,3,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[4,3,1,6,2,7,8,5] => [1,6,2,7,8,3,4,5] => [5,4,3,8,7,2,6,1] => ? => ? = 1 + 1
[7,8,1,5,6,2,3,4] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[2,7,8,1,6,3,4,5] => [1,6,2,7,8,3,4,5] => [5,4,3,8,7,2,6,1] => ? => ? = 1 + 1
[2,5,4,1,6,7,8,3] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[6,3,5,1,2,7,8,4] => [1,2,7,8,3,5,4,6] => [6,4,5,3,8,7,2,1] => ? => ? = 1 + 1
[8,5,4,1,6,2,3,7] => [1,6,2,3,7,4,5,8] => [8,5,4,7,3,2,6,1] => ? => ? = 1 + 1
[9,1,2,3,4,7,8,5,6] => [1,2,3,4,7,8,5,6,9] => [9,6,5,8,7,4,3,2,1] => ? => ? = 1 + 1
[5,6,7,8,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[5,6,7,9,4,3,2,10,8,1] => [1,2,10,3,4,5,6,7,9,8] => [8,9,7,6,5,4,3,10,2,1] => ? => ? = 1 + 1
[4,5,6,3,2,1,9,10,8,7] => [1,9,10,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,10,9,1] => [1,1,1,1,1,1,2,1,1] => ? = 1 + 1
[3,4,2,1,8,9,10,7,6,5] => [1,8,9,10,2,3,4,5,6,7] => [7,6,5,4,3,2,10,9,8,1] => ? => ? = 1 + 1
[2,1,7,8,9,10,6,5,4,3] => [1,7,8,9,10,2,3,4,5,6] => [6,5,4,3,2,10,9,8,7,1] => ? => ? = 1 + 1
Description
The largest part of an integer composition.
Mp00223: Permutations runsortPermutations
Mp00064: Permutations reversePermutations
Mp00071: Permutations descent compositionInteger compositions
St000760: Integer compositions ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,3,1] => [2,1] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,3,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 2 = 1 + 1
[8,7,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[7,8,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[7,8,2,3,4,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[8,6,5,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[8,5,6,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[7,4,3,2,1,5,6,8] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[1,2,4,7,8,6,5,3] => [1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ? = 1 + 1
[3,2,1,5,6,8,7,4] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[2,4,3,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[3,2,4,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[3,2,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[2,3,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[3,2,4,1,5,8,7,6] => [1,5,8,2,4,3,6,7] => [7,6,3,4,2,8,5,1] => ? => ? = 1 + 1
[4,3,2,1,5,6,8,7] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[2,3,4,1,5,6,8,7] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[3,2,4,1,7,6,5,8] => [1,7,2,4,3,5,8,6] => [6,8,5,3,4,2,7,1] => ? => ? = 1 + 1
[3,2,4,1,5,7,6,8] => [1,5,7,2,4,3,6,8] => [8,6,3,4,2,7,5,1] => ? => ? = 1 + 1
[3,2,5,4,1,6,7,8] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[1,3,8,5,4,6,7,2] => [1,3,8,2,4,6,7,5] => [5,7,6,4,2,8,3,1] => ? => ? = 1 + 1
[1,7,5,4,3,6,8,2] => [1,7,2,3,6,8,4,5] => [5,4,8,6,3,2,7,1] => ? => ? = 1 + 1
[5,2,4,3,1,6,7,8] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[2,4,1,6,7,8,3,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[2,5,1,6,7,8,3,4] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[2,1,6,3,7,4,5,8] => [1,6,2,3,7,4,5,8] => [8,5,4,7,3,2,6,1] => ? => ? = 1 + 1
[2,3,6,1,4,7,5,8] => [1,4,7,2,3,6,5,8] => [8,5,6,3,2,7,4,1] => ? => ? = 1 + 1
[3,1,4,6,8,2,5,7] => [1,4,6,8,2,5,7,3] => [3,7,5,2,8,6,4,1] => ? => ? = 1 + 1
[4,6,1,2,7,8,3,5] => [1,2,7,8,3,5,4,6] => [6,4,5,3,8,7,2,1] => ? => ? = 1 + 1
[4,1,2,6,7,3,8,5] => [1,2,6,7,3,8,4,5] => [5,4,8,3,7,6,2,1] => ? => ? = 1 + 1
[1,2,4,5,7,3,8,6] => [1,2,4,5,7,3,8,6] => [6,8,3,7,5,4,2,1] => ? => ? = 1 + 1
[2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,2,4,1] => ? => ? = 1 + 1
[8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[9,7,5,10,3,8,2,6,1,4] => [1,4,2,6,3,8,5,10,7,9] => [9,7,10,5,8,3,6,2,4,1] => [1,2,2,2,2,1] => ? = 1 + 1
[4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[6,3,5,1,2,4,7,8] => [1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ? = 1 + 1
[8,3,5,1,2,4,6,7] => [1,2,4,6,7,3,5,8] => [8,5,3,7,6,4,2,1] => ? => ? = 1 + 1
[8,1,4,7,6,2,3,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[4,3,1,6,2,7,8,5] => [1,6,2,7,8,3,4,5] => [5,4,3,8,7,2,6,1] => ? => ? = 1 + 1
[7,8,1,5,6,2,3,4] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[2,7,8,1,6,3,4,5] => [1,6,2,7,8,3,4,5] => [5,4,3,8,7,2,6,1] => ? => ? = 1 + 1
[2,5,4,1,6,7,8,3] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[6,3,5,1,2,7,8,4] => [1,2,7,8,3,5,4,6] => [6,4,5,3,8,7,2,1] => ? => ? = 1 + 1
[8,5,4,1,6,2,3,7] => [1,6,2,3,7,4,5,8] => [8,5,4,7,3,2,6,1] => ? => ? = 1 + 1
[9,1,2,3,4,7,8,5,6] => [1,2,3,4,7,8,5,6,9] => [9,6,5,8,7,4,3,2,1] => ? => ? = 1 + 1
[5,6,7,8,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[5,6,7,9,4,3,2,10,8,1] => [1,2,10,3,4,5,6,7,9,8] => [8,9,7,6,5,4,3,10,2,1] => ? => ? = 1 + 1
[4,5,6,3,2,1,9,10,8,7] => [1,9,10,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,10,9,1] => [1,1,1,1,1,1,2,1,1] => ? = 1 + 1
[3,4,2,1,8,9,10,7,6,5] => [1,8,9,10,2,3,4,5,6,7] => [7,6,5,4,3,2,10,9,8,1] => ? => ? = 1 + 1
[2,1,7,8,9,10,6,5,4,3] => [1,7,8,9,10,2,3,4,5,6] => [6,5,4,3,2,10,9,8,7,1] => ? => ? = 1 + 1
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition. By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
Matching statistic: St000764
Mp00223: Permutations runsortPermutations
Mp00069: Permutations complementPermutations
Mp00071: Permutations descent compositionInteger compositions
St000764: Integer compositions ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [1,1,2] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [1,1,2] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 2 = 1 + 1
[8,6,5,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[8,5,6,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[4,2,3,5,6,1,7,8] => [1,7,8,2,3,5,6,4] => [8,2,1,7,6,4,3,5] => ? => ? = 1 + 1
[5,4,6,3,1,2,7,8] => [1,2,7,8,3,4,6,5] => [8,7,2,1,6,5,3,4] => ? => ? = 1 + 1
[4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[1,2,3,7,8,6,5,4] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[2,4,3,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [8,3,2,1,7,5,6,4] => ? => ? = 1 + 1
[3,2,4,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [8,3,2,1,7,5,6,4] => ? => ? = 1 + 1
[3,2,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[2,3,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[3,2,4,1,5,8,7,6] => [1,5,8,2,4,3,6,7] => [8,4,1,7,5,6,3,2] => ? => ? = 1 + 1
[2,4,3,1,6,5,8,7] => [1,6,2,4,3,5,8,7] => [8,3,7,5,6,4,1,2] => ? => ? = 1 + 1
[3,2,4,1,6,5,8,7] => [1,6,2,4,3,5,8,7] => [8,3,7,5,6,4,1,2] => ? => ? = 1 + 1
[3,2,4,1,5,7,6,8] => [1,5,7,2,4,3,6,8] => [8,4,2,7,5,6,3,1] => ? => ? = 1 + 1
[1,7,5,4,3,6,8,2] => [1,7,2,3,6,8,4,5] => [8,2,7,6,3,1,5,4] => ? => ? = 1 + 1
[5,2,4,3,1,6,7,8] => [1,6,7,8,2,4,3,5] => [8,3,2,1,7,5,6,4] => ? => ? = 1 + 1
[2,4,1,3,6,5,8,7] => [1,3,6,2,4,5,8,7] => [8,6,3,7,5,4,1,2] => ? => ? = 1 + 1
[2,4,1,6,3,5,8,7] => [1,6,2,4,3,5,8,7] => [8,3,7,5,6,4,1,2] => ? => ? = 1 + 1
[2,4,6,1,8,3,5,7] => [1,8,2,4,6,3,5,7] => [8,1,7,5,3,6,4,2] => ? => ? = 1 + 1
[2,4,1,6,7,8,3,5] => [1,6,7,8,2,4,3,5] => [8,3,2,1,7,5,6,4] => ? => ? = 1 + 1
[2,3,5,6,1,7,8,4] => [1,7,8,2,3,5,6,4] => [8,2,1,7,6,4,3,5] => ? => ? = 1 + 1
[2,3,6,1,4,7,5,8] => [1,4,7,2,3,6,5,8] => [8,5,2,7,6,3,4,1] => ? => ? = 1 + 1
[3,4,6,1,2,7,8,5] => [1,2,7,8,3,4,6,5] => [8,7,2,1,6,5,3,4] => ? => ? = 1 + 1
[4,5,1,2,7,3,6,8] => [1,2,7,3,6,8,4,5] => [8,7,2,6,3,1,5,4] => ? => ? = 1 + 1
[4,1,7,8,2,3,5,6] => [1,7,8,2,3,5,6,4] => [8,2,1,7,6,4,3,5] => ? => ? = 1 + 1
[4,1,2,3,7,8,5,6] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[5,6,1,2,3,7,8,4] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[5,7,1,2,8,3,4,6] => [1,2,8,3,4,6,5,7] => [8,7,1,6,5,3,4,2] => ? => ? = 1 + 1
[6,1,2,3,7,8,4,5] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => [10,7,9,8,5,6,3,4,1,2] => ? => ? = 1 + 1
[8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[9,7,5,10,3,8,2,6,1,4] => [1,4,2,6,3,8,5,10,7,9] => [10,7,9,5,8,3,6,1,4,2] => [1,2,2,2,2,1] => ? = 1 + 1
[4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[7,5,8,2,4,1,3,6] => [1,3,6,2,4,5,8,7] => [8,6,3,7,5,4,1,2] => ? => ? = 1 + 1
[8,1,4,7,6,2,3,5] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[8,3,4,1,7,2,5,6] => [1,7,2,5,6,3,4,8] => [8,2,7,4,3,6,5,1] => ? => ? = 1 + 1
[7,3,4,6,1,2,8,5] => [1,2,8,3,4,6,5,7] => [8,7,1,6,5,3,4,2] => ? => ? = 1 + 1
[2,3,8,7,6,1,4,5] => [1,4,5,2,3,8,6,7] => [8,5,4,7,6,1,3,2] => ? => ? = 1 + 1
[9,1,2,3,4,7,8,5,6] => [1,2,3,4,7,8,5,6,9] => [9,8,7,6,3,2,5,4,1] => ? => ? = 1 + 1
[5,6,7,8,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[5,6,7,9,4,3,2,10,8,1] => [1,2,10,3,4,5,6,7,9,8] => [10,9,1,8,7,6,5,4,2,3] => ? => ? = 1 + 1
[4,5,6,3,2,1,9,10,8,7] => [1,9,10,2,3,4,5,6,7,8] => [10,2,1,9,8,7,6,5,4,3] => [1,1,2,1,1,1,1,1,1] => ? = 1 + 1
[3,4,2,1,8,9,10,7,6,5] => [1,8,9,10,2,3,4,5,6,7] => [10,3,2,1,9,8,7,6,5,4] => [1,1,1,2,1,1,1,1,1] => ? = 1 + 1
[2,1,7,8,9,10,6,5,4,3] => [1,7,8,9,10,2,3,4,5,6] => [10,4,3,2,1,9,8,7,6,5] => [1,1,1,1,2,1,1,1,1] => ? = 1 + 1
[4,5,3,7,8,6,1,2] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[4,2,5,6,8,1,7,3] => [1,7,2,5,6,8,3,4] => [8,2,7,4,3,1,6,5] => ? => ? = 1 + 1
[5,4,3,7,8,6,1,2] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[5,4,2,6,7,1,3,8] => [1,3,8,2,6,7,4,5] => [8,6,1,7,3,2,5,4] => ? => ? = 1 + 1
Description
The number of strong records in an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Mp00223: Permutations runsortPermutations
Mp00064: Permutations reversePermutations
Mp00071: Permutations descent compositionInteger compositions
St000903: Integer compositions ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,3,1] => [2,1] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,3,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => [2,1,1] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [1,2,1] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => [2,1,1] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 2 = 1 + 1
[8,7,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[7,8,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[7,8,2,3,4,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[8,6,5,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[8,5,6,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[7,4,3,2,1,5,6,8] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[1,2,4,7,8,6,5,3] => [1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ? = 1 + 1
[3,2,1,5,6,8,7,4] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[2,4,3,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[3,2,4,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[3,2,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[2,3,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[3,2,4,1,5,8,7,6] => [1,5,8,2,4,3,6,7] => [7,6,3,4,2,8,5,1] => ? => ? = 1 + 1
[4,3,2,1,5,6,8,7] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[2,3,4,1,5,6,8,7] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? = 1 + 1
[3,2,4,1,7,6,5,8] => [1,7,2,4,3,5,8,6] => [6,8,5,3,4,2,7,1] => ? => ? = 1 + 1
[3,2,4,1,5,7,6,8] => [1,5,7,2,4,3,6,8] => [8,6,3,4,2,7,5,1] => ? => ? = 1 + 1
[3,2,5,4,1,6,7,8] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[1,3,8,5,4,6,7,2] => [1,3,8,2,4,6,7,5] => [5,7,6,4,2,8,3,1] => ? => ? = 1 + 1
[1,7,5,4,3,6,8,2] => [1,7,2,3,6,8,4,5] => [5,4,8,6,3,2,7,1] => ? => ? = 1 + 1
[5,2,4,3,1,6,7,8] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[2,4,1,6,7,8,3,5] => [1,6,7,8,2,4,3,5] => [5,3,4,2,8,7,6,1] => ? => ? = 1 + 1
[2,5,1,6,7,8,3,4] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[2,1,6,3,7,4,5,8] => [1,6,2,3,7,4,5,8] => [8,5,4,7,3,2,6,1] => ? => ? = 1 + 1
[2,3,6,1,4,7,5,8] => [1,4,7,2,3,6,5,8] => [8,5,6,3,2,7,4,1] => ? => ? = 1 + 1
[3,1,4,6,8,2,5,7] => [1,4,6,8,2,5,7,3] => [3,7,5,2,8,6,4,1] => ? => ? = 1 + 1
[4,6,1,2,7,8,3,5] => [1,2,7,8,3,5,4,6] => [6,4,5,3,8,7,2,1] => ? => ? = 1 + 1
[4,1,2,6,7,3,8,5] => [1,2,6,7,3,8,4,5] => [5,4,8,3,7,6,2,1] => ? => ? = 1 + 1
[1,2,4,5,7,3,8,6] => [1,2,4,5,7,3,8,6] => [6,8,3,7,5,4,2,1] => ? => ? = 1 + 1
[2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,2,4,1] => ? => ? = 1 + 1
[8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[9,7,5,10,3,8,2,6,1,4] => [1,4,2,6,3,8,5,10,7,9] => [9,7,10,5,8,3,6,2,4,1] => [1,2,2,2,2,1] => ? = 1 + 1
[4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[6,3,5,1,2,4,7,8] => [1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ? = 1 + 1
[8,3,5,1,2,4,6,7] => [1,2,4,6,7,3,5,8] => [8,5,3,7,6,4,2,1] => ? => ? = 1 + 1
[8,1,4,7,6,2,3,5] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? = 1 + 1
[4,3,1,6,2,7,8,5] => [1,6,2,7,8,3,4,5] => [5,4,3,8,7,2,6,1] => ? => ? = 1 + 1
[7,8,1,5,6,2,3,4] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? = 1 + 1
[2,7,8,1,6,3,4,5] => [1,6,2,7,8,3,4,5] => [5,4,3,8,7,2,6,1] => ? => ? = 1 + 1
[2,5,4,1,6,7,8,3] => [1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ? = 1 + 1
[6,3,5,1,2,7,8,4] => [1,2,7,8,3,5,4,6] => [6,4,5,3,8,7,2,1] => ? => ? = 1 + 1
[8,5,4,1,6,2,3,7] => [1,6,2,3,7,4,5,8] => [8,5,4,7,3,2,6,1] => ? => ? = 1 + 1
[9,1,2,3,4,7,8,5,6] => [1,2,3,4,7,8,5,6,9] => [9,6,5,8,7,4,3,2,1] => ? => ? = 1 + 1
[5,6,7,8,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 1 + 1
[5,6,7,9,4,3,2,10,8,1] => [1,2,10,3,4,5,6,7,9,8] => [8,9,7,6,5,4,3,10,2,1] => ? => ? = 1 + 1
[4,5,6,3,2,1,9,10,8,7] => [1,9,10,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,10,9,1] => [1,1,1,1,1,1,2,1,1] => ? = 1 + 1
Description
The number of different parts of an integer composition.
Mp00223: Permutations runsortPermutations
Mp00069: Permutations complementPermutations
Mp00071: Permutations descent compositionInteger compositions
St000758: Integer compositions ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [1,1,2] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [1,1,2] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [1,2,1] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,2,1] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 2 = 1 + 1
[8,6,5,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[8,5,6,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[4,2,3,5,6,1,7,8] => [1,7,8,2,3,5,6,4] => [8,2,1,7,6,4,3,5] => ? => ? = 1 + 1
[5,4,6,3,1,2,7,8] => [1,2,7,8,3,4,6,5] => [8,7,2,1,6,5,3,4] => ? => ? = 1 + 1
[4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[1,2,3,7,8,6,5,4] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[2,4,3,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [8,3,2,1,7,5,6,4] => ? => ? = 1 + 1
[3,2,4,1,6,7,8,5] => [1,6,7,8,2,4,3,5] => [8,3,2,1,7,5,6,4] => ? => ? = 1 + 1
[3,2,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[2,3,1,4,7,6,8,5] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[3,2,4,1,5,8,7,6] => [1,5,8,2,4,3,6,7] => [8,4,1,7,5,6,3,2] => ? => ? = 1 + 1
[2,4,3,1,6,5,8,7] => [1,6,2,4,3,5,8,7] => [8,3,7,5,6,4,1,2] => ? => ? = 1 + 1
[3,2,4,1,6,5,8,7] => [1,6,2,4,3,5,8,7] => [8,3,7,5,6,4,1,2] => ? => ? = 1 + 1
[3,2,4,1,5,7,6,8] => [1,5,7,2,4,3,6,8] => [8,4,2,7,5,6,3,1] => ? => ? = 1 + 1
[1,7,5,4,3,6,8,2] => [1,7,2,3,6,8,4,5] => [8,2,7,6,3,1,5,4] => ? => ? = 1 + 1
[5,2,4,3,1,6,7,8] => [1,6,7,8,2,4,3,5] => [8,3,2,1,7,5,6,4] => ? => ? = 1 + 1
[2,4,1,3,6,5,8,7] => [1,3,6,2,4,5,8,7] => [8,6,3,7,5,4,1,2] => ? => ? = 1 + 1
[2,4,1,6,3,5,8,7] => [1,6,2,4,3,5,8,7] => [8,3,7,5,6,4,1,2] => ? => ? = 1 + 1
[2,4,6,1,8,3,5,7] => [1,8,2,4,6,3,5,7] => [8,1,7,5,3,6,4,2] => ? => ? = 1 + 1
[2,4,1,6,7,8,3,5] => [1,6,7,8,2,4,3,5] => [8,3,2,1,7,5,6,4] => ? => ? = 1 + 1
[2,3,5,6,1,7,8,4] => [1,7,8,2,3,5,6,4] => [8,2,1,7,6,4,3,5] => ? => ? = 1 + 1
[2,3,6,1,4,7,5,8] => [1,4,7,2,3,6,5,8] => [8,5,2,7,6,3,4,1] => ? => ? = 1 + 1
[3,4,6,1,2,7,8,5] => [1,2,7,8,3,4,6,5] => [8,7,2,1,6,5,3,4] => ? => ? = 1 + 1
[4,5,1,2,7,3,6,8] => [1,2,7,3,6,8,4,5] => [8,7,2,6,3,1,5,4] => ? => ? = 1 + 1
[4,1,7,8,2,3,5,6] => [1,7,8,2,3,5,6,4] => [8,2,1,7,6,4,3,5] => ? => ? = 1 + 1
[4,1,2,3,7,8,5,6] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[5,6,1,2,3,7,8,4] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[5,7,1,2,8,3,4,6] => [1,2,8,3,4,6,5,7] => [8,7,1,6,5,3,4,2] => ? => ? = 1 + 1
[6,1,2,3,7,8,4,5] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
[2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => [10,7,9,8,5,6,3,4,1,2] => ? => ? = 1 + 1
[8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[9,7,5,10,3,8,2,6,1,4] => [1,4,2,6,3,8,5,10,7,9] => [10,7,9,5,8,3,6,1,4,2] => [1,2,2,2,2,1] => ? = 1 + 1
[4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[7,5,8,2,4,1,3,6] => [1,3,6,2,4,5,8,7] => [8,6,3,7,5,4,1,2] => ? => ? = 1 + 1
[8,1,4,7,6,2,3,5] => [1,4,7,2,3,5,6,8] => [8,5,2,7,6,4,3,1] => ? => ? = 1 + 1
[8,3,4,1,7,2,5,6] => [1,7,2,5,6,3,4,8] => [8,2,7,4,3,6,5,1] => ? => ? = 1 + 1
[7,3,4,6,1,2,8,5] => [1,2,8,3,4,6,5,7] => [8,7,1,6,5,3,4,2] => ? => ? = 1 + 1
[2,3,8,7,6,1,4,5] => [1,4,5,2,3,8,6,7] => [8,5,4,7,6,1,3,2] => ? => ? = 1 + 1
[9,1,2,3,4,7,8,5,6] => [1,2,3,4,7,8,5,6,9] => [9,8,7,6,3,2,5,4,1] => ? => ? = 1 + 1
[5,6,7,8,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[5,6,7,9,4,3,2,10,8,1] => [1,2,10,3,4,5,6,7,9,8] => [10,9,1,8,7,6,5,4,2,3] => ? => ? = 1 + 1
[4,5,6,3,2,1,9,10,8,7] => [1,9,10,2,3,4,5,6,7,8] => [10,2,1,9,8,7,6,5,4,3] => [1,1,2,1,1,1,1,1,1] => ? = 1 + 1
[3,4,2,1,8,9,10,7,6,5] => [1,8,9,10,2,3,4,5,6,7] => [10,3,2,1,9,8,7,6,5,4] => [1,1,1,2,1,1,1,1,1] => ? = 1 + 1
[2,1,7,8,9,10,6,5,4,3] => [1,7,8,9,10,2,3,4,5,6] => [10,4,3,2,1,9,8,7,6,5] => [1,1,1,1,2,1,1,1,1] => ? = 1 + 1
[6,7,8,9,10,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[4,5,3,7,8,6,1,2] => [1,2,3,7,8,4,5,6] => [8,7,6,2,1,5,4,3] => ? => ? = 1 + 1
Description
The length of the longest staircase fitting into an integer composition. For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Mp00223: Permutations runsortPermutations
Mp00204: Permutations LLPSInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001035: Dyck paths ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [1,1]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1,2] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,2,1] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,8,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,5,6,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,4,5,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,4,5,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,6,7,4,5,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,5,6,7,8,3,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,5,3,4,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,5,3,4,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,6,7,5,3,4,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,5,6,3,4,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,3,4,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,8,3,4,5,2,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,5,4,2,3,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,5,4,2,3,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,6,7,5,4,2,3,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,5,6,4,2,3,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,4,2,3,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,4,5,2,3,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,6,7,4,5,2,3,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,5,6,7,8,2,3,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,5,2,3,4,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,8,5,2,3,4,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,5,6,7,2,3,4,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,2,3,4,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,5,4,3,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,6,7,5,4,3,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,5,6,4,3,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,4,3,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,4,5,3,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,4,5,3,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,5,6,7,8,3,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,5,3,4,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,3,4,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,3,4,5,6,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,3,4,5,6,7,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[3,4,5,6,7,8,1,2] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,5,4,1,2,3] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,8,5,4,1,2,3] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,5,6,7,4,1,2,3] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,4,1,2,3] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,4,5,1,2,3] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,6,7,4,5,1,2,3] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,4,5,6,1,2,3] => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path. A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino. For example, any rotation of a Ferrers shape has convexity degree at most one. The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St000442
Mp00223: Permutations runsortPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St000442: Dyck paths ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[8,6,4,3,2,1,5,7] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[8,6,3,4,2,1,5,7] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[5,3,1,2,4,6,7,8] => [1,2,4,6,7,8,3,5] => ?
=> ?
=> ? = 1
[2,1,4,5,8,7,6,3] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[1,2,4,6,7,8,5,3] => [1,2,4,6,7,8,3,5] => ?
=> ?
=> ? = 1
[3,2,1,5,7,6,8,4] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[2,1,3,4,6,7,8,5] => [1,3,4,6,7,8,2,5] => ?
=> ?
=> ? = 1
[3,2,1,4,5,8,7,6] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[2,1,4,5,6,3,8,7] => [1,4,5,6,2,3,8,7] => ?
=> ?
=> ? = 1
[1,3,4,5,7,6,2,8] => [1,3,4,5,7,2,8,6] => ?
=> ?
=> ? = 1
[3,2,4,1,6,7,5,8] => [1,6,7,2,4,3,5,8] => ?
=> ?
=> ? = 1
[4,3,2,1,5,7,6,8] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[2,3,4,1,5,7,6,8] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[2,1,4,5,8,6,7,3] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[2,4,6,1,8,3,5,7] => [1,8,2,4,6,3,5,7] => ?
=> ?
=> ? = 1
[2,1,4,5,8,3,6,7] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[2,5,1,3,7,8,4,6] => [1,3,7,8,2,5,4,6] => ?
=> ?
=> ? = 1
[2,5,7,1,8,3,4,6] => [1,8,2,5,7,3,4,6] => ?
=> ?
=> ? = 1
[2,3,6,1,4,5,8,7] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[3,5,1,7,8,2,4,6] => [1,7,8,2,4,6,3,5] => ?
=> ?
=> ? = 1
[3,6,1,7,2,4,8,5] => [1,7,2,4,8,3,6,5] => ?
=> ?
=> ? = 1
[1,3,6,2,4,7,8,5] => [1,3,6,2,4,7,8,5] => ?
=> ?
=> ? = 1
[3,1,2,4,6,7,8,5] => [1,2,4,6,7,8,3,5] => ?
=> ?
=> ? = 1
[1,2,4,6,7,8,3,5] => [1,2,4,6,7,8,3,5] => ?
=> ?
=> ? = 1
[1,2,5,3,6,7,4,8] => [1,2,5,3,6,7,4,8] => ?
=> ?
=> ? = 1
[2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => [.,[[.,.],[.,[[.,.],[[.,.],[[.,.],.]]]]]]
=> ?
=> ? = 1
[8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ?
=> ? = 1
[6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ?
=> ? = 1
[9,7,5,10,3,8,2,6,1,4] => [1,4,2,6,3,8,5,10,7,9] => [.,[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ?
=> ? = 1
[2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ?
=> ? = 1
[10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,2,8,1,3,4,5,7] => [1,3,4,5,7,2,8,6] => ?
=> ?
=> ? = 1
[8,3,5,1,2,4,6,7] => [1,2,4,6,7,3,5,8] => ?
=> ?
=> ? = 1
[4,2,6,8,1,3,5,7] => [1,3,5,7,2,6,8,4] => ?
=> ?
=> ? = 1
[5,4,2,6,1,3,7,8] => [1,3,7,8,2,6,4,5] => ?
=> ?
=> ? = 1
[7,4,2,5,1,3,6,8] => [1,3,6,8,2,5,4,7] => ?
=> ?
=> ? = 1
[8,4,7,2,6,1,3,5] => [1,3,5,2,6,4,7,8] => ?
=> ?
=> ? = 1
[6,5,3,2,7,1,4,8] => [1,4,8,2,7,3,5,6] => ?
=> ?
=> ? = 1
[8,7,4,2,6,1,3,5] => [1,3,5,2,6,4,7,8] => ?
=> ?
=> ? = 1
[7,1,4,5,6,2,8,3] => [1,4,5,6,2,8,3,7] => ?
=> ?
=> ? = 1
[8,4,1,2,6,7,3,5] => [1,2,6,7,3,5,4,8] => ?
=> ?
=> ? = 1
[2,6,4,1,3,7,8,5] => [1,3,7,8,2,6,4,5] => ?
=> ?
=> ? = 1
[2,6,5,1,3,7,8,4] => [1,3,7,8,2,6,4,5] => ?
=> ?
=> ? = 1
[8,3,5,1,6,7,2,4] => [1,6,7,2,4,3,5,8] => ?
=> ?
=> ? = 1
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,1,2,3,4,5,6,9,7] => [1,2,3,4,5,6,9,7,8] => [.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[7,1,2,3,4,5,8,9,6] => [1,2,3,4,5,8,9,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]]
=> ?
=> ? = 1
[9,1,2,3,4,7,8,5,6] => [1,2,3,4,7,8,5,6,9] => [.,[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]]
=> ?
=> ? = 1
Description
The maximal area to the right of an up step of a Dyck path.
Mp00223: Permutations runsortPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St001031: Dyck paths ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[8,6,4,3,2,1,5,7] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[8,6,3,4,2,1,5,7] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[5,3,1,2,4,6,7,8] => [1,2,4,6,7,8,3,5] => ?
=> ?
=> ? = 1
[2,1,4,5,8,7,6,3] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[1,2,4,6,7,8,5,3] => [1,2,4,6,7,8,3,5] => ?
=> ?
=> ? = 1
[3,2,1,5,7,6,8,4] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[2,1,3,4,6,7,8,5] => [1,3,4,6,7,8,2,5] => ?
=> ?
=> ? = 1
[3,2,1,4,5,8,7,6] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[2,1,4,5,6,3,8,7] => [1,4,5,6,2,3,8,7] => ?
=> ?
=> ? = 1
[1,3,4,5,7,6,2,8] => [1,3,4,5,7,2,8,6] => ?
=> ?
=> ? = 1
[3,2,4,1,6,7,5,8] => [1,6,7,2,4,3,5,8] => ?
=> ?
=> ? = 1
[4,3,2,1,5,7,6,8] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[2,3,4,1,5,7,6,8] => [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 1
[2,1,4,5,8,6,7,3] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[2,4,6,1,8,3,5,7] => [1,8,2,4,6,3,5,7] => ?
=> ?
=> ? = 1
[2,1,4,5,8,3,6,7] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[2,5,1,3,7,8,4,6] => [1,3,7,8,2,5,4,6] => ?
=> ?
=> ? = 1
[2,5,7,1,8,3,4,6] => [1,8,2,5,7,3,4,6] => ?
=> ?
=> ? = 1
[2,3,6,1,4,5,8,7] => [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 1
[3,5,1,7,8,2,4,6] => [1,7,8,2,4,6,3,5] => ?
=> ?
=> ? = 1
[3,6,1,7,2,4,8,5] => [1,7,2,4,8,3,6,5] => ?
=> ?
=> ? = 1
[1,3,6,2,4,7,8,5] => [1,3,6,2,4,7,8,5] => ?
=> ?
=> ? = 1
[3,1,2,4,6,7,8,5] => [1,2,4,6,7,8,3,5] => ?
=> ?
=> ? = 1
[1,2,4,6,7,8,3,5] => [1,2,4,6,7,8,3,5] => ?
=> ?
=> ? = 1
[1,2,5,3,6,7,4,8] => [1,2,5,3,6,7,4,8] => ?
=> ?
=> ? = 1
[2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => [.,[[.,.],[.,[[.,.],[[.,.],[[.,.],.]]]]]]
=> ?
=> ? = 1
[8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ?
=> ? = 1
[6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ?
=> ? = 1
[9,7,5,10,3,8,2,6,1,4] => [1,4,2,6,3,8,5,10,7,9] => [.,[[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ?
=> ? = 1
[2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => [.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ?
=> ? = 1
[10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,2,8,1,3,4,5,7] => [1,3,4,5,7,2,8,6] => ?
=> ?
=> ? = 1
[8,3,5,1,2,4,6,7] => [1,2,4,6,7,3,5,8] => ?
=> ?
=> ? = 1
[4,2,6,8,1,3,5,7] => [1,3,5,7,2,6,8,4] => ?
=> ?
=> ? = 1
[5,4,2,6,1,3,7,8] => [1,3,7,8,2,6,4,5] => ?
=> ?
=> ? = 1
[7,4,2,5,1,3,6,8] => [1,3,6,8,2,5,4,7] => ?
=> ?
=> ? = 1
[8,4,7,2,6,1,3,5] => [1,3,5,2,6,4,7,8] => ?
=> ?
=> ? = 1
[6,5,3,2,7,1,4,8] => [1,4,8,2,7,3,5,6] => ?
=> ?
=> ? = 1
[8,7,4,2,6,1,3,5] => [1,3,5,2,6,4,7,8] => ?
=> ?
=> ? = 1
[7,1,4,5,6,2,8,3] => [1,4,5,6,2,8,3,7] => ?
=> ?
=> ? = 1
[8,4,1,2,6,7,3,5] => [1,2,6,7,3,5,4,8] => ?
=> ?
=> ? = 1
[2,6,4,1,3,7,8,5] => [1,3,7,8,2,6,4,5] => ?
=> ?
=> ? = 1
[2,6,5,1,3,7,8,4] => [1,3,7,8,2,6,4,5] => ?
=> ?
=> ? = 1
[8,3,5,1,6,7,2,4] => [1,6,7,2,4,3,5,8] => ?
=> ?
=> ? = 1
[9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,1,2,3,4,5,6,9,7] => [1,2,3,4,5,6,9,7,8] => [.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[7,1,2,3,4,5,8,9,6] => [1,2,3,4,5,8,9,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]]
=> ?
=> ? = 1
[9,1,2,3,4,7,8,5,6] => [1,2,3,4,7,8,5,6,9] => [.,[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]]
=> ?
=> ? = 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
The following 126 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000444The length of the maximal rise of a Dyck path. St001732The number of peaks visible from the left. St000659The number of rises of length at least 2 of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000480The number of lower covers of a partition in dominance order. St001092The number of distinct even parts of a partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000668The least common multiple of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001128The exponens consonantiae of a partition. St001432The order dimension of the partition. St000759The smallest missing part in an integer partition. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000783The side length of the largest staircase partition fitting into a partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000396The register function (or Horton-Strahler number) of a binary tree. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001203We associate to 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 Dyck path as follows: St000013The height of a Dyck path. St001335The cardinality of a minimal cycle-isolating set of a graph. St000253The crossing number of a set partition. St000701The protection number of a binary tree. St000451The length of the longest pattern of the form k 1 2. St001111The weak 2-dynamic chromatic number of a graph. St000535The rank-width of a graph. St000552The number of cut vertices of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001743The discrepancy of a graph. St000259The diameter of a connected graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000397The Strahler number of a rooted tree. St000679The pruning number of an ordered tree. St000846The maximal number of elements covering an element of a poset. St001062The maximal size of a block of a set partition. St001261The Castelnuovo-Mumford regularity of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000264The girth of a graph, which is not a tree. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000058The order of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000094The depth of an ordered tree. St000485The length of the longest cycle of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St000456The monochromatic index of a connected 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 $eA$. St001665The number of pure excedances of a permutation. St000862The number of parts of the shifted shape of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000308The height of the tree associated to a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000387The matching number of a graph. St000486The number of cycles of length at least 3 of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001271The competition number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001728The number of invisible descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000328The maximum number of child nodes in a tree. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001734The lettericity of a graph. St001330The hat guessing number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000454The largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001488The number of corners of a skew partition. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001896The number of right descents of a signed permutations. St001624The breadth of a lattice. St000699The toughness times the least common multiple of 1,. St000640The rank of the largest boolean interval in a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation.