Your data matches 85 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000504
(load all 2 compositions to match this statistic)
Mapping
St000504: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => 2
{{1},{2}}
 => 1
{{1,2,3}}
 => 3
{{1,2},{3}}
 => 2
{{1,3},{2}}
 => 2
{{1},{2,3}}
 => 1
{{1},{2},{3}}
 => 1
{{1,2,3,4}}
 => 4
{{1,2,3},{4}}
 => 3
{{1,2,4},{3}}
 => 3
{{1,2},{3,4}}
 => 2
{{1,2},{3},{4}}
 => 2
{{1,3,4},{2}}
 => 3
{{1,3},{2,4}}
 => 2
{{1,3},{2},{4}}
 => 2
{{1,4},{2,3}}
 => 2
{{1},{2,3,4}}
 => 1
{{1},{2,3},{4}}
 => 1
{{1,4},{2},{3}}
 => 2
{{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => 1
{{1},{2},{3},{4}}
 => 1
{{1,2,3,4,5}}
 => 5
{{1,2,3,4},{5}}
 => 4
{{1,2,3,5},{4}}
 => 4
{{1,2,3},{4,5}}
 => 3
{{1,2,3},{4},{5}}
 => 3
{{1,2,4,5},{3}}
 => 4
{{1,2,4},{3,5}}
 => 3
{{1,2,4},{3},{5}}
 => 3
{{1,2,5},{3,4}}
 => 3
{{1,2},{3,4,5}}
 => 2
{{1,2},{3,4},{5}}
 => 2
{{1,2,5},{3},{4}}
 => 3
{{1,2},{3,5},{4}}
 => 2
{{1,2},{3},{4,5}}
 => 2
{{1,2},{3},{4},{5}}
 => 2
{{1,3,4,5},{2}}
 => 4
{{1,3,4},{2,5}}
 => 3
{{1,3,4},{2},{5}}
 => 3
{{1,3,5},{2,4}}
 => 3
{{1,3},{2,4,5}}
 => 2
{{1,3},{2,4},{5}}
 => 2
{{1,3,5},{2},{4}}
 => 3
{{1,3},{2,5},{4}}
 => 2
{{1,3},{2},{4,5}}
 => 2
{{1,3},{2},{4},{5}}
 => 2
{{1,4,5},{2,3}}
 => 3
{{1,4},{2,3,5}}
 => 2
{{1,4},{2,3},{5}}
 => 2
Description
The cardinality of the first block of a set partition. The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St000502
(load all 2 compositions to match this statistic)
Mapping
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => 1 = 2 - 1
{{1},{2}}
 => 0 = 1 - 1
{{1,2,3}}
 => 2 = 3 - 1
{{1,2},{3}}
 => 1 = 2 - 1
{{1,3},{2}}
 => 0 = 1 - 1
{{1},{2,3}}
 => 1 = 2 - 1
{{1},{2},{3}}
 => 0 = 1 - 1
{{1,2,3,4}}
 => 3 = 4 - 1
{{1,2,3},{4}}
 => 2 = 3 - 1
{{1,2,4},{3}}
 => 1 = 2 - 1
{{1,2},{3,4}}
 => 2 = 3 - 1
{{1,2},{3},{4}}
 => 1 = 2 - 1
{{1,3,4},{2}}
 => 1 = 2 - 1
{{1,3},{2,4}}
 => 0 = 1 - 1
{{1,3},{2},{4}}
 => 0 = 1 - 1
{{1,4},{2,3}}
 => 1 = 2 - 1
{{1},{2,3,4}}
 => 2 = 3 - 1
{{1},{2,3},{4}}
 => 1 = 2 - 1
{{1,4},{2},{3}}
 => 0 = 1 - 1
{{1},{2,4},{3}}
 => 0 = 1 - 1
{{1},{2},{3,4}}
 => 1 = 2 - 1
{{1},{2},{3},{4}}
 => 0 = 1 - 1
{{1,2,3,4,5}}
 => 4 = 5 - 1
{{1,2,3,4},{5}}
 => 3 = 4 - 1
{{1,2,3,5},{4}}
 => 2 = 3 - 1
{{1,2,3},{4,5}}
 => 3 = 4 - 1
{{1,2,3},{4},{5}}
 => 2 = 3 - 1
{{1,2,4,5},{3}}
 => 2 = 3 - 1
{{1,2,4},{3,5}}
 => 1 = 2 - 1
{{1,2,4},{3},{5}}
 => 1 = 2 - 1
{{1,2,5},{3,4}}
 => 2 = 3 - 1
{{1,2},{3,4,5}}
 => 3 = 4 - 1
{{1,2},{3,4},{5}}
 => 2 = 3 - 1
{{1,2,5},{3},{4}}
 => 1 = 2 - 1
{{1,2},{3,5},{4}}
 => 1 = 2 - 1
{{1,2},{3},{4,5}}
 => 2 = 3 - 1
{{1,2},{3},{4},{5}}
 => 1 = 2 - 1
{{1,3,4,5},{2}}
 => 2 = 3 - 1
{{1,3,4},{2,5}}
 => 1 = 2 - 1
{{1,3,4},{2},{5}}
 => 1 = 2 - 1
{{1,3,5},{2,4}}
 => 0 = 1 - 1
{{1,3},{2,4,5}}
 => 1 = 2 - 1
{{1,3},{2,4},{5}}
 => 0 = 1 - 1
{{1,3,5},{2},{4}}
 => 0 = 1 - 1
{{1,3},{2,5},{4}}
 => 0 = 1 - 1
{{1,3},{2},{4,5}}
 => 1 = 2 - 1
{{1,3},{2},{4},{5}}
 => 0 = 1 - 1
{{1,4,5},{2,3}}
 => 2 = 3 - 1
{{1,4},{2,3,5}}
 => 1 = 2 - 1
{{1,4},{2,3},{5}}
 => 1 = 2 - 1
Description
The number of successions of a set partitions. This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => 2
{{1},{2}}
 => [1,1] => 1
{{1,2,3}}
 => [3] => 3
{{1,2},{3}}
 => [2,1] => 2
{{1,3},{2}}
 => [2,1] => 2
{{1},{2,3}}
 => [1,2] => 1
{{1},{2},{3}}
 => [1,1,1] => 1
{{1,2,3,4}}
 => [4] => 4
{{1,2,3},{4}}
 => [3,1] => 3
{{1,2,4},{3}}
 => [3,1] => 3
{{1,2},{3,4}}
 => [2,2] => 2
{{1,2},{3},{4}}
 => [2,1,1] => 2
{{1,3,4},{2}}
 => [3,1] => 3
{{1,3},{2,4}}
 => [2,2] => 2
{{1,3},{2},{4}}
 => [2,1,1] => 2
{{1,4},{2,3}}
 => [2,2] => 2
{{1},{2,3,4}}
 => [1,3] => 1
{{1},{2,3},{4}}
 => [1,2,1] => 1
{{1,4},{2},{3}}
 => [2,1,1] => 2
{{1},{2,4},{3}}
 => [1,2,1] => 1
{{1},{2},{3,4}}
 => [1,1,2] => 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => 1
{{1,2,3,4,5}}
 => [5] => 5
{{1,2,3,4},{5}}
 => [4,1] => 4
{{1,2,3,5},{4}}
 => [4,1] => 4
{{1,2,3},{4,5}}
 => [3,2] => 3
{{1,2,3},{4},{5}}
 => [3,1,1] => 3
{{1,2,4,5},{3}}
 => [4,1] => 4
{{1,2,4},{3,5}}
 => [3,2] => 3
{{1,2,4},{3},{5}}
 => [3,1,1] => 3
{{1,2,5},{3,4}}
 => [3,2] => 3
{{1,2},{3,4,5}}
 => [2,3] => 2
{{1,2},{3,4},{5}}
 => [2,2,1] => 2
{{1,2,5},{3},{4}}
 => [3,1,1] => 3
{{1,2},{3,5},{4}}
 => [2,2,1] => 2
{{1,2},{3},{4,5}}
 => [2,1,2] => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => 2
{{1,3,4,5},{2}}
 => [4,1] => 4
{{1,3,4},{2,5}}
 => [3,2] => 3
{{1,3,4},{2},{5}}
 => [3,1,1] => 3
{{1,3,5},{2,4}}
 => [3,2] => 3
{{1,3},{2,4,5}}
 => [2,3] => 2
{{1,3},{2,4},{5}}
 => [2,2,1] => 2
{{1,3,5},{2},{4}}
 => [3,1,1] => 3
{{1,3},{2,5},{4}}
 => [2,2,1] => 2
{{1,3},{2},{4,5}}
 => [2,1,2] => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => 2
{{1,4,5},{2,3}}
 => [3,2] => 3
{{1,4},{2,3,5}}
 => [2,3] => 2
{{1,4},{2,3},{5}}
 => [2,2,1] => 2
Description
The first part of an integer composition.
Matching statistic: St000025
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1,0,0]
 => 2
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 2
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 2
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 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: St000026
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1,0,0]
 => 2
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 2
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 2
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
Description
The position of the first return of a Dyck path.
Matching statistic: St000383
(load all 6 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [2] => 2
{{1},{2}}
 => [1,1] => [1,1] => 1
{{1,2,3}}
 => [3] => [3] => 3
{{1,2},{3}}
 => [2,1] => [1,2] => 2
{{1,3},{2}}
 => [2,1] => [1,2] => 2
{{1},{2,3}}
 => [1,2] => [2,1] => 1
{{1},{2},{3}}
 => [1,1,1] => [1,1,1] => 1
{{1,2,3,4}}
 => [4] => [4] => 4
{{1,2,3},{4}}
 => [3,1] => [1,3] => 3
{{1,2,4},{3}}
 => [3,1] => [1,3] => 3
{{1,2},{3,4}}
 => [2,2] => [2,2] => 2
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,2] => 2
{{1,3,4},{2}}
 => [3,1] => [1,3] => 3
{{1,3},{2,4}}
 => [2,2] => [2,2] => 2
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,2] => 2
{{1,4},{2,3}}
 => [2,2] => [2,2] => 2
{{1},{2,3,4}}
 => [1,3] => [3,1] => 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,2,1] => 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,2] => 2
{{1},{2,4},{3}}
 => [1,2,1] => [1,2,1] => 1
{{1},{2},{3,4}}
 => [1,1,2] => [2,1,1] => 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,1,1,1] => 1
{{1,2,3,4,5}}
 => [5] => [5] => 5
{{1,2,3,4},{5}}
 => [4,1] => [1,4] => 4
{{1,2,3,5},{4}}
 => [4,1] => [1,4] => 4
{{1,2,3},{4,5}}
 => [3,2] => [2,3] => 3
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => 3
{{1,2,4,5},{3}}
 => [4,1] => [1,4] => 4
{{1,2,4},{3,5}}
 => [3,2] => [2,3] => 3
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => 3
{{1,2,5},{3,4}}
 => [3,2] => [2,3] => 3
{{1,2},{3,4,5}}
 => [2,3] => [3,2] => 2
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => 2
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => 3
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => 2
{{1,2},{3},{4,5}}
 => [2,1,2] => [2,1,2] => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,1,2] => 2
{{1,3,4,5},{2}}
 => [4,1] => [1,4] => 4
{{1,3,4},{2,5}}
 => [3,2] => [2,3] => 3
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => 3
{{1,3,5},{2,4}}
 => [3,2] => [2,3] => 3
{{1,3},{2,4,5}}
 => [2,3] => [3,2] => 2
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => 2
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => 3
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => 2
{{1,3},{2},{4,5}}
 => [2,1,2] => [2,1,2] => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,1,2] => 2
{{1,4,5},{2,3}}
 => [3,2] => [2,3] => 3
{{1,4},{2,3,5}}
 => [2,3] => [3,2] => 2
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => 2
Description
The last part of an integer composition.
Matching statistic: St000733
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [[1,2]]
 => [[1],[2]]
 => 2
{{1},{2}}
 => [[1],[2]]
 => [[1,2]]
 => 1
{{1,2,3}}
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
{{1,2},{3}}
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
{{1},{2,3}}
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 3
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 2
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 1
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 2
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 1
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 2
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 4
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 2
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => 4
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 2
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => 1
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 1
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => 3
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => 2
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => 2
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 4
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => 2
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => 1
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 3
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 3
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => 1
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 3
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 2
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 2
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => 1
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 3
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 3
{{1,4},{2,3},{5}}
 => [[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1,0,0]
 => 3 = 2 + 1
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
{{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1] => 11 => 2
{{1},{2}}
 => [1,1] => [2] => 10 => 1
{{1,2,3}}
 => [3] => [1,1,1] => 111 => 3
{{1,2},{3}}
 => [2,1] => [1,2] => 110 => 2
{{1,3},{2}}
 => [2,1] => [1,2] => 110 => 2
{{1},{2,3}}
 => [1,2] => [2,1] => 101 => 1
{{1},{2},{3}}
 => [1,1,1] => [3] => 100 => 1
{{1,2,3,4}}
 => [4] => [1,1,1,1] => 1111 => 4
{{1,2,3},{4}}
 => [3,1] => [1,1,2] => 1110 => 3
{{1,2,4},{3}}
 => [3,1] => [1,1,2] => 1110 => 3
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => 1101 => 2
{{1,2},{3},{4}}
 => [2,1,1] => [1,3] => 1100 => 2
{{1,3,4},{2}}
 => [3,1] => [1,1,2] => 1110 => 3
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => 1101 => 2
{{1,3},{2},{4}}
 => [2,1,1] => [1,3] => 1100 => 2
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => 1101 => 2
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => 1011 => 1
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 1010 => 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,3] => 1100 => 2
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 1010 => 1
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => 1001 => 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 1000 => 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 11111 => 5
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => 11110 => 4
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,2] => 11110 => 4
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => 11101 => 3
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => 11100 => 3
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,2] => 11110 => 4
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => 11101 => 3
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => 11100 => 3
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => 11101 => 3
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => 11011 => 2
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => 11010 => 2
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => 11100 => 3
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => 11010 => 2
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 11001 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => 11000 => 2
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,2] => 11110 => 4
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => 11101 => 3
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => 11100 => 3
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => 11101 => 3
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => 11011 => 2
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => 11010 => 2
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => 11100 => 3
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => 11010 => 2
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => 11001 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,4] => 11000 => 2
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => 11101 => 3
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => 11011 => 2
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => 11010 => 2
Description
The number of leading ones in a binary word.
The following 75 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000505The biggest entry in the block containing the 1. St000678The number of up steps after the last double rise of a Dyck path. St000823The number of unsplittable factors of the set partition. St000971The smallest closer of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000445The number of rises of length 1 of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001484The number of singletons of an integer partition. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000363The number of minimal vertex covers of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001316The domatic number of a graph. St000069The number of maximal elements of a poset. St000617The number of global maxima of a Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000068The number of minimal elements in a poset. St000260The radius of a connected graph. St000441The number of successions of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000054The first entry of the permutation. St000989The number of final rises of a permutation. 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. St000007The number of saliances of the permutation. St000237The number of small exceedances. St000031The number of cycles in the cycle decomposition of a permutation. St000214The number of adjacencies of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001948The number of augmented double ascents of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000654The first descent of a permutation. St000338The number of pixed points of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000942The number of critical left to right maxima of the parking functions.