Your data matches 112 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000159
(load all 2 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => 1
[[1,2]]
 => [2]
 => 1
[[1],[2]]
 => [1,1]
 => 1
[[1,2,3]]
 => [3]
 => 1
[[1,3],[2]]
 => [2,1]
 => 2
[[1,2],[3]]
 => [2,1]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => 1
[[1,3,4],[2]]
 => [3,1]
 => 2
[[1,2,4],[3]]
 => [3,1]
 => 2
[[1,2,3],[4]]
 => [3,1]
 => 2
[[1,3],[2,4]]
 => [2,2]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => 2
[[1,3,5],[2,4]]
 => [3,2]
 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => 2
[[1,4],[2,5],[3]]
 => [2,2,1]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => 3
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
(load all 2 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => 2 = 1 + 1
[[1,2]]
 => [2]
 => 2 = 1 + 1
[[1],[2]]
 => [1,1]
 => 2 = 1 + 1
[[1,2,3]]
 => [3]
 => 2 = 1 + 1
[[1,3],[2]]
 => [2,1]
 => 3 = 2 + 1
[[1,2],[3]]
 => [2,1]
 => 3 = 2 + 1
[[1],[2],[3]]
 => [1,1,1]
 => 2 = 1 + 1
[[1,3,4],[2]]
 => [3,1]
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [3,1]
 => 3 = 2 + 1
[[1,3],[2,4]]
 => [2,2]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => [2,2]
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [2,1,1]
 => 3 = 2 + 1
[[1,3],[2],[4]]
 => [2,1,1]
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => [2,1,1]
 => 3 = 2 + 1
[[1,3,5],[2,4]]
 => [3,2]
 => 3 = 2 + 1
[[1,2,5],[3,4]]
 => [3,2]
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => [3,2]
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,2]
 => 3 = 2 + 1
[[1,2,3],[4,5]]
 => [3,2]
 => 3 = 2 + 1
[[1,4,5],[2],[3]]
 => [3,1,1]
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => [3,1,1]
 => 3 = 2 + 1
[[1,2,5],[3],[4]]
 => [3,1,1]
 => 3 = 2 + 1
[[1,3,4],[2],[5]]
 => [3,1,1]
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => [3,1,1]
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => [3,1,1]
 => 3 = 2 + 1
[[1,4],[2,5],[3]]
 => [2,2,1]
 => 3 = 2 + 1
[[1,3],[2,5],[4]]
 => [2,2,1]
 => 3 = 2 + 1
[[1,2],[3,5],[4]]
 => [2,2,1]
 => 3 = 2 + 1
[[1,3],[2,4],[5]]
 => [2,2,1]
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => [2,2,1]
 => 3 = 2 + 1
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => 4 = 3 + 1
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => 4 = 3 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000053
(load all 6 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => [1,0,1,0]
 => 1
[[1,2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1],[2]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[[1,2,3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,3],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,2],[3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,3,4],[2]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,2,3],[4]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,3],[2,4]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,3,5],[2,4]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
Description
The number of valleys of the Dyck path.
Matching statistic: St000291
(load all 7 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => 10 => 1
[[1,2]]
 => [2]
 => 100 => 1
[[1],[2]]
 => [1,1]
 => 110 => 1
[[1,2,3]]
 => [3]
 => 1000 => 1
[[1,3],[2]]
 => [2,1]
 => 1010 => 2
[[1,2],[3]]
 => [2,1]
 => 1010 => 2
[[1],[2],[3]]
 => [1,1,1]
 => 1110 => 1
[[1,3,4],[2]]
 => [3,1]
 => 10010 => 2
[[1,2,4],[3]]
 => [3,1]
 => 10010 => 2
[[1,2,3],[4]]
 => [3,1]
 => 10010 => 2
[[1,3],[2,4]]
 => [2,2]
 => 1100 => 1
[[1,2],[3,4]]
 => [2,2]
 => 1100 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => 10110 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => 10110 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => 10110 => 2
[[1,3,5],[2,4]]
 => [3,2]
 => 10100 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => 10100 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => 10100 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => 10100 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => 10100 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => 100110 => 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => 100110 => 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => 100110 => 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => 100110 => 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => 100110 => 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => 100110 => 2
[[1,4],[2,5],[3]]
 => [2,2,1]
 => 11010 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => 11010 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => 11010 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => 11010 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => 11010 => 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => 101010 => 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => 101010 => 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => 101010 => 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => 101010 => 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => 101010 => 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => 101010 => 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => 101010 => 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => 101010 => 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => 101010 => 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => 101010 => 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => 101010 => 3
Description
The number of descents of a binary word.
Matching statistic: St000306
(load all 3 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => [1,0,1,0]
 => 1
[[1,2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1],[2]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[[1,2,3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,3],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,2],[3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,3,4],[2]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,2,3],[4]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,3],[2,4]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,3,5],[2,4]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000390
(load all 7 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => 10 => 1
[[1,2]]
 => [2]
 => 100 => 1
[[1],[2]]
 => [1,1]
 => 110 => 1
[[1,2,3]]
 => [3]
 => 1000 => 1
[[1,3],[2]]
 => [2,1]
 => 1010 => 2
[[1,2],[3]]
 => [2,1]
 => 1010 => 2
[[1],[2],[3]]
 => [1,1,1]
 => 1110 => 1
[[1,3,4],[2]]
 => [3,1]
 => 10010 => 2
[[1,2,4],[3]]
 => [3,1]
 => 10010 => 2
[[1,2,3],[4]]
 => [3,1]
 => 10010 => 2
[[1,3],[2,4]]
 => [2,2]
 => 1100 => 1
[[1,2],[3,4]]
 => [2,2]
 => 1100 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => 10110 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => 10110 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => 10110 => 2
[[1,3,5],[2,4]]
 => [3,2]
 => 10100 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => 10100 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => 10100 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => 10100 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => 10100 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => 100110 => 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => 100110 => 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => 100110 => 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => 100110 => 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => 100110 => 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => 100110 => 2
[[1,4],[2,5],[3]]
 => [2,2,1]
 => 11010 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => 11010 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => 11010 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => 11010 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => 11010 => 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => 101010 => 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => 101010 => 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => 101010 => 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => 101010 => 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => 101010 => 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => 101010 => 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => 101010 => 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => 101010 => 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => 101010 => 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => 101010 => 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => 101010 => 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => 101010 => 3
Description
The number of runs of ones in a binary word.
Matching statistic: St000764
(load all 2 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000764: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => 1
[[1,2]]
 => [1,2] => [2] => 1
[[1],[2]]
 => [2,1] => [1,1] => 1
[[1,2,3]]
 => [1,2,3] => [3] => 1
[[1,3],[2]]
 => [2,1,3] => [1,2] => 2
[[1,2],[3]]
 => [3,1,2] => [1,2] => 2
[[1],[2],[3]]
 => [3,2,1] => [1,1,1] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [1,3] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,3] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [1,3] => 2
[[1,3],[2,4]]
 => [2,4,1,3] => [2,2] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,2] => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,2] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,2] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => 2
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,3] => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [2,3] => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,3] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,3] => 2
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,3] => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,1,3] => 2
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,1,3] => 2
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,1,3] => 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,1,3] => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,3] => 2
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 2
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,2,2] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,2,2] => 2
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,2] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,2,2] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,2] => 2
[[1,4,6],[2,5],[3]]
 => [3,2,5,1,4,6] => [1,2,3] => 3
[[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [1,2,3] => 3
[[1,2,6],[3,5],[4]]
 => [4,3,5,1,2,6] => [1,2,3] => 3
[[1,3,6],[2,4],[5]]
 => [5,2,4,1,3,6] => [1,2,3] => 3
[[1,2,6],[3,4],[5]]
 => [5,3,4,1,2,6] => [1,2,3] => 3
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [1,2,3] => 3
[[1,3,5],[2,6],[4]]
 => [4,2,6,1,3,5] => [1,2,3] => 3
[[1,2,5],[3,6],[4]]
 => [4,3,6,1,2,5] => [1,2,3] => 3
[[1,3,4],[2,6],[5]]
 => [5,2,6,1,3,4] => [1,2,3] => 3
[[1,2,4],[3,6],[5]]
 => [5,3,6,1,2,4] => [1,2,3] => 3
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [1,2,3] => 3
[[1,3,5],[2,4],[6]]
 => [6,2,4,1,3,5] => [1,2,3] => 3
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => [1,2,3] => 3
[[1,3,4],[2,5],[6]]
 => [6,2,5,1,3,4] => [1,2,3] => 3
[[1,2,4],[3,5],[6]]
 => [6,3,5,1,2,4] => [1,2,3] => 3
[[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [1,2,3] => 3
Description
The number of strong records in an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Matching statistic: St000767
(load all 4 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => 1
[[1,2]]
 => [1,2] => [2] => 1
[[1],[2]]
 => [2,1] => [1,1] => 1
[[1,2,3]]
 => [1,2,3] => [3] => 1
[[1,3],[2]]
 => [2,1,3] => [1,2] => 2
[[1,2],[3]]
 => [3,1,2] => [1,2] => 2
[[1],[2],[3]]
 => [3,2,1] => [1,1,1] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [1,3] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,3] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [1,3] => 2
[[1,3],[2,4]]
 => [2,4,1,3] => [2,2] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,2] => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,2] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,2] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => 2
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,3] => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [2,3] => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,3] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,3] => 2
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,3] => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,1,3] => 2
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,1,3] => 2
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,1,3] => 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,1,3] => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,3] => 2
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 2
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,2,2] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,2,2] => 2
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,2] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,2,2] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,2] => 2
[[1,4,6],[2,5],[3]]
 => [3,2,5,1,4,6] => [1,2,3] => 3
[[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [1,2,3] => 3
[[1,2,6],[3,5],[4]]
 => [4,3,5,1,2,6] => [1,2,3] => 3
[[1,3,6],[2,4],[5]]
 => [5,2,4,1,3,6] => [1,2,3] => 3
[[1,2,6],[3,4],[5]]
 => [5,3,4,1,2,6] => [1,2,3] => 3
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [1,2,3] => 3
[[1,3,5],[2,6],[4]]
 => [4,2,6,1,3,5] => [1,2,3] => 3
[[1,2,5],[3,6],[4]]
 => [4,3,6,1,2,5] => [1,2,3] => 3
[[1,3,4],[2,6],[5]]
 => [5,2,6,1,3,4] => [1,2,3] => 3
[[1,2,4],[3,6],[5]]
 => [5,3,6,1,2,4] => [1,2,3] => 3
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [1,2,3] => 3
[[1,3,5],[2,4],[6]]
 => [6,2,4,1,3,5] => [1,2,3] => 3
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => [1,2,3] => 3
[[1,3,4],[2,5],[6]]
 => [6,2,5,1,3,4] => [1,2,3] => 3
[[1,2,4],[3,5],[6]]
 => [6,3,5,1,2,4] => [1,2,3] => 3
[[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [1,2,3] => 3
Description
The number of runs in an integer composition. Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1]. It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].
Matching statistic: St000903
(load all 9 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => 1
[[1,2]]
 => [1,2] => [2] => 1
[[1],[2]]
 => [2,1] => [1,1] => 1
[[1,2,3]]
 => [1,2,3] => [3] => 1
[[1,3],[2]]
 => [2,1,3] => [1,2] => 2
[[1,2],[3]]
 => [3,1,2] => [1,2] => 2
[[1],[2],[3]]
 => [3,2,1] => [1,1,1] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [1,3] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,3] => 2
[[1,2,3],[4]]
 => [4,1,2,3] => [1,3] => 2
[[1,3],[2,4]]
 => [2,4,1,3] => [2,2] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [2,2] => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,2] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,2] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,2] => 2
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,3] => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [2,3] => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,3] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,3] => 2
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,3] => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,1,3] => 2
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,1,3] => 2
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,1,3] => 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,1,3] => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,3] => 2
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,3] => 2
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,2,2] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,2,2] => 2
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,2] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,2,2] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,2] => 2
[[1,4,6],[2,5],[3]]
 => [3,2,5,1,4,6] => [1,2,3] => 3
[[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [1,2,3] => 3
[[1,2,6],[3,5],[4]]
 => [4,3,5,1,2,6] => [1,2,3] => 3
[[1,3,6],[2,4],[5]]
 => [5,2,4,1,3,6] => [1,2,3] => 3
[[1,2,6],[3,4],[5]]
 => [5,3,4,1,2,6] => [1,2,3] => 3
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [1,2,3] => 3
[[1,3,5],[2,6],[4]]
 => [4,2,6,1,3,5] => [1,2,3] => 3
[[1,2,5],[3,6],[4]]
 => [4,3,6,1,2,5] => [1,2,3] => 3
[[1,3,4],[2,6],[5]]
 => [5,2,6,1,3,4] => [1,2,3] => 3
[[1,2,4],[3,6],[5]]
 => [5,3,6,1,2,4] => [1,2,3] => 3
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [1,2,3] => 3
[[1,3,5],[2,4],[6]]
 => [6,2,4,1,3,5] => [1,2,3] => 3
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => [1,2,3] => 3
[[1,3,4],[2,5],[6]]
 => [6,2,5,1,3,4] => [1,2,3] => 3
[[1,2,4],[3,5],[6]]
 => [6,3,5,1,2,4] => [1,2,3] => 3
[[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [1,2,3] => 3
Description
The number of different parts of an integer composition.
Matching statistic: St001142
(load all 5 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => [1,0,1,0]
 => 1
[[1,2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[[1],[2]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[[1,2,3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,3],[2]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1,2],[3]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[[1],[2],[3]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,3,4],[2]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,2,3],[4]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[[1,3],[2,4]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,3,5],[2,4]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,5],[3,4]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,3,4],[2,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,3],[4,5]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 2
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,4],[3,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,2,3],[4,5],[6]]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 3
Description
The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The following 102 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000015The number of peaks of a Dyck path. St000292The number of ascents of a binary word. St000761The number of ascents in an integer composition. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. 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: St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000021The number of descents of a permutation. 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. St000069The number of maximal elements of a poset. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000245The number of ascents of a permutation. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000522The number of 1-protected nodes of a rooted tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000996The number of exclusive left-to-right maxima of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001424The number of distinct squares in a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. St001726The number of visible inversions of a permutation. St001732The number of peaks visible from the left. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000647The number of big descents of a permutation. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000925The number of topologically connected components of a set partition. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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 $eA$. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001673The degree of asymmetry of an integer composition. St001840The number of descents of a set partition. St001960The number of descents of a permutation minus one if its first entry is not one. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000568The hook number of a binary tree. St000646The number of big ascents of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001487The number of inner corners of a skew partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.