searching the database
Your data matches 106 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: St000470
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => 3
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => 3
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => 2
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 0 = 1 - 1
[1,2] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[2,1] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,3,2] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[2,1,3] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[2,3,1] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[3,2,1] => [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,4,3,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[2,1,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[4,1,3,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[4,2,3,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [2,1] => [2]
=> 1
[2,1] => [1,2] => [2,1] => [2]
=> 1
[1,2,3] => [1,2,3] => [3,2,1] => [3]
=> 1
[1,3,2] => [1,2,3] => [3,2,1] => [3]
=> 1
[2,1,3] => [1,2,3] => [3,2,1] => [3]
=> 1
[2,3,1] => [1,2,3] => [3,2,1] => [3]
=> 1
[3,1,2] => [1,3,2] => [2,3,1] => [2,1]
=> 2
[3,2,1] => [1,3,2] => [2,3,1] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4]
=> 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [4]
=> 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [4]
=> 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [4]
=> 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [3,1]
=> 2
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [3,1]
=> 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [4]
=> 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [4]
=> 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [4]
=> 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4]
=> 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [3,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [3,1]
=> 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [3,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [3,1]
=> 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [3,1]
=> 2
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [3,1]
=> 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [3,1]
=> 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [3,1]
=> 2
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 3
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [3,1]
=> 2
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 3
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [3,1]
=> 2
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [3,1]
=> 2
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [3,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [5]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [4,1]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [4,1]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [4,1]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [4,1]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [4,1]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [4,1]
=> 2
Description
The length of the partition.
Matching statistic: St000011
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,1,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,4,3,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,1,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,4,3,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,1,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,4,1,2] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[4,1,3,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[4,2,3,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,2,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 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: St000025
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[3,1,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,4,2,3] => [1,2,4,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,4,3,2] => [1,2,4,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[2,4,1,3] => [1,2,4,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[3,1,2,4] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[3,2,1,4] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[3,4,1,2] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[3,4,2,1] => [1,3,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,1,2,3] => [1,4,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,1,3,2] => [1,4,2,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,4,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,2,3,1] => [1,4,2,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,4,2,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,2,1] => [1,4,2,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 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: St000097
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [2] => ([],2)
=> 1
[2,1] => [1,2] => [2] => ([],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [1,2,3] => [3] => ([],3)
=> 1
[2,1,3] => [1,2,3] => [3] => ([],3)
=> 1
[2,3,1] => [1,2,3] => [3] => ([],3)
=> 1
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,4,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,4,1,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[3,2,1,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,2,4,1] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[3,4,1,2] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,4,2,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,3,1,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,3,2,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [2] => ([],2)
=> 1
[2,1] => [1,2] => [2] => ([],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [1,2,3] => [3] => ([],3)
=> 1
[2,1,3] => [1,2,3] => [3] => ([],3)
=> 1
[2,3,1] => [1,2,3] => [3] => ([],3)
=> 1
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,4,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,4,1,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[3,2,1,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,2,4,1] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[3,4,1,2] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,4,2,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,3,1,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,3,2,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000167
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [[]]
=> 1
[1,2] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[2,1] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St000172
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [2] => ([],2)
=> 1
[2,1] => [1,2] => [2] => ([],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [1,2,3] => [3] => ([],3)
=> 1
[2,1,3] => [1,2,3] => [3] => ([],3)
=> 1
[2,3,1] => [1,2,3] => [3] => ([],3)
=> 1
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,4,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,4,1,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[3,2,1,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,2,4,1] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[3,4,1,2] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,4,2,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,3,1,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,3,2,1] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000288
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 => 1
[1,2] => [1,2] => [2] => 10 => 1
[2,1] => [1,2] => [2] => 10 => 1
[1,2,3] => [1,2,3] => [3] => 100 => 1
[1,3,2] => [1,2,3] => [3] => 100 => 1
[2,1,3] => [1,2,3] => [3] => 100 => 1
[2,3,1] => [1,2,3] => [3] => 100 => 1
[3,1,2] => [1,3,2] => [2,1] => 101 => 2
[3,2,1] => [1,3,2] => [2,1] => 101 => 2
[1,2,3,4] => [1,2,3,4] => [4] => 1000 => 1
[1,2,4,3] => [1,2,3,4] => [4] => 1000 => 1
[1,3,2,4] => [1,2,3,4] => [4] => 1000 => 1
[1,3,4,2] => [1,2,3,4] => [4] => 1000 => 1
[1,4,2,3] => [1,2,4,3] => [3,1] => 1001 => 2
[1,4,3,2] => [1,2,4,3] => [3,1] => 1001 => 2
[2,1,3,4] => [1,2,3,4] => [4] => 1000 => 1
[2,1,4,3] => [1,2,3,4] => [4] => 1000 => 1
[2,3,1,4] => [1,2,3,4] => [4] => 1000 => 1
[2,3,4,1] => [1,2,3,4] => [4] => 1000 => 1
[2,4,1,3] => [1,2,4,3] => [3,1] => 1001 => 2
[2,4,3,1] => [1,2,4,3] => [3,1] => 1001 => 2
[3,1,2,4] => [1,3,2,4] => [2,2] => 1010 => 2
[3,1,4,2] => [1,3,4,2] => [3,1] => 1001 => 2
[3,2,1,4] => [1,3,2,4] => [2,2] => 1010 => 2
[3,2,4,1] => [1,3,4,2] => [3,1] => 1001 => 2
[3,4,1,2] => [1,3,2,4] => [2,2] => 1010 => 2
[3,4,2,1] => [1,3,2,4] => [2,2] => 1010 => 2
[4,1,2,3] => [1,4,3,2] => [2,1,1] => 1011 => 3
[4,1,3,2] => [1,4,2,3] => [2,2] => 1010 => 2
[4,2,1,3] => [1,4,3,2] => [2,1,1] => 1011 => 3
[4,2,3,1] => [1,4,2,3] => [2,2] => 1010 => 2
[4,3,1,2] => [1,4,2,3] => [2,2] => 1010 => 2
[4,3,2,1] => [1,4,2,3] => [2,2] => 1010 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => 10000 => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => 10000 => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => 10000 => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => 10000 => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => 10001 => 2
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => 10001 => 2
[1,3,2,4,5] => [1,2,3,4,5] => [5] => 10000 => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => 10000 => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => 10000 => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => 10000 => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => 10001 => 2
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => 10001 => 2
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => 10010 => 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => 10001 => 2
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => 10010 => 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => 10001 => 2
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => 10010 => 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 96 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000507The number of ascents of a standard tableau. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000439The position of the first down step of a Dyck path. St000536The pathwidth of a graph. St000662The staircase size of the code of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001176The size of a partition minus its first part. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000619The number of cyclic descents of a permutation. St000054The first entry of the permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000354The number of recoils of a permutation. St000703The number of deficiencies of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000306The bounce count of a Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000740The last entry of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001489The maximum of the number of descents and the number of inverse descents. St001427The number of descents of a signed permutation. St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St001187The number of simple modules with grade at least one 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000051The size of the left subtree of a binary tree. St000168The number of internal nodes of an ordered tree. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001812The biclique partition number of a graph. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000711The number of big exceedences of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001330The hat guessing number of a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001960The number of descents of a permutation minus one if its first entry is not one. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001624The breadth of a lattice. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in 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!