Your data matches 168 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000147
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => 2 = 1 + 1
([(2,3)],4)
 => [6,6]
 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 7 = 6 + 1
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => [2,2]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [4,4]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [8,8]
 => 2 = 1 + 1
([(2,3)],4)
 => [6,6]
 => [2,2,2,2,2,2]
 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3,3]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 7 = 6 + 1
Description
The length of the partition.
Matching statistic: St000734
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => 2 = 1 + 1
([(2,3)],4)
 => [6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7 = 6 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000013
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7 = 6 + 1
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000643
(load all 2 compositions to match this statistic)
Mapping
St000643: Posets ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ? = 1 + 1
([],2)
 => 2 = 1 + 1
([(0,1)],2)
 => 3 = 2 + 1
([],3)
 => 2 = 1 + 1
([(1,2)],3)
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => 3 = 2 + 1
([],4)
 => 2 = 1 + 1
([(2,3)],4)
 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => 7 = 6 + 1
Description
The size of the largest orbit of antichains under Panyushev complementation.
Matching statistic: St000474
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => ? = 1 + 1
([(2,3)],4)
 => [6,6]
 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 7 = 6 + 1
Description
Dyson's crank of a partition. Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as $$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
Matching statistic: St000676
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 1 + 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => 2 = 1 + 1
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 1 + 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1]
 => [[1],[2]]
 => 1
([],2)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2
([],3)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 5
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
([],4)
 => [2,2,2,2,2,2,2,2]
 => [8,8]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
 => ? = 1
([(2,3)],4)
 => [6,6]
 => [2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => 5
([(1,2),(1,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9],[10]]
 => 5
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 3
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 3
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 3
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9],[10]]
 => 5
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => 2
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => 2
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 4
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6
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: St000288
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => 100 => 011 => 2 = 1 + 1
([],2)
 => [2,2]
 => 1100 => 0011 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => 1000 => 0111 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => 111100 => 000011 => 2 = 1 + 1
([(1,2)],3)
 => [6]
 => 1000000 => 0111111 => 6 = 5 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 10100 => 01011 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => 10000 => 01111 => 4 = 3 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 10100 => 01011 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => 1111111100 => 0000000011 => ? = 1 + 1
([(2,3)],4)
 => [6,6]
 => 11000000 => 00111111 => 6 = 5 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => 100001100 => 011110011 => 6 = 5 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 1011100 => 0100011 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 10000000 => 01111111 => 7 = 6 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 100100 => 011011 => 4 = 3 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => 110000 => 001111 => 4 = 3 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 100100 => 011011 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => 100001100 => 011110011 => 6 = 5 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 100100 => 011011 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 1011100 => 0100011 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => 111000 => 000111 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 1001000 => 0110111 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 101100 => 010011 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 100000 => 011111 => 5 = 4 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 10000000 => 01111111 => 7 = 6 + 1
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
The following 158 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000439The position of the first down step of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000025The number of initial rises of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and 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. St001480The number of simple summands of the module J^2/J^3. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001645The pebbling number of a connected graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000738The first entry in the last row of a standard tableau. St000144The pyramid weight of the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000259The diameter of a connected graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000456The monochromatic index of a connected graph. St000468The Hosoya index of a graph. St000537The cutwidth of a graph. St000553The number of blocks of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001271The competition number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000271The chromatic index of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000469The distinguishing number of a graph. St000552The number of cut vertices of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001366The maximal multiplicity of a degree of a vertex of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000907The number of maximal antichains of minimal length in a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000455The second largest eigenvalue of a graph if it is integral. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000906The length of the shortest maximal chain in a poset. St001664The number of non-isomorphic subposets of a poset. St000086The number of subgraphs. St000450The number of edges minus the number of vertices plus 2 of a graph. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St000315The number of isolated vertices of a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in 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 $eA$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001330The hat guessing number of a graph. St000993The multiplicity of the largest part of an integer partition. St001118The acyclic chromatic index of a 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 $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001624The breadth of a lattice. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000264The girth of a graph, which is not a tree. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St001875The number of simple modules with projective dimension at most 1. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.