searching the database
Your data matches 92 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001453
St001453: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1 = 2 - 1
[[1],[1]]
=> 1 = 2 - 1
[[2]]
=> 1 = 2 - 1
[[1,1]]
=> 1 = 2 - 1
[[1],[1],[1]]
=> 1 = 2 - 1
[[2],[1]]
=> 2 = 3 - 1
[[1,1],[1]]
=> 1 = 2 - 1
[[3]]
=> 1 = 2 - 1
[[2,1]]
=> 2 = 3 - 1
[[1,1,1]]
=> 1 = 2 - 1
[[1],[1],[1],[1]]
=> 1 = 2 - 1
[[2],[1],[1]]
=> 2 = 3 - 1
[[2],[2]]
=> 1 = 2 - 1
[[1,1],[1],[1]]
=> 1 = 2 - 1
[[1,1],[1,1]]
=> 1 = 2 - 1
[[3],[1]]
=> 2 = 3 - 1
[[2,1],[1]]
=> 2 = 3 - 1
[[1,1,1],[1]]
=> 1 = 2 - 1
[[4]]
=> 1 = 2 - 1
[[3,1]]
=> 2 = 3 - 1
[[2,2]]
=> 1 = 2 - 1
[[2,1,1]]
=> 2 = 3 - 1
[[1,1,1,1]]
=> 1 = 2 - 1
[[1],[1],[1],[1],[1]]
=> 1 = 2 - 1
[[2],[1],[1],[1]]
=> 2 = 3 - 1
[[2],[2],[1]]
=> 2 = 3 - 1
[[1,1],[1],[1],[1]]
=> 1 = 2 - 1
[[1,1],[1,1],[1]]
=> 1 = 2 - 1
[[3],[1],[1]]
=> 2 = 3 - 1
[[3],[2]]
=> 2 = 3 - 1
[[2,1],[1],[1]]
=> 2 = 3 - 1
[[2,1],[2]]
=> 2 = 3 - 1
[[2,1],[1,1]]
=> 2 = 3 - 1
[[1,1,1],[1],[1]]
=> 1 = 2 - 1
[[1,1,1],[1,1]]
=> 1 = 2 - 1
[[4],[1]]
=> 2 = 3 - 1
[[3,1],[1]]
=> 2 = 3 - 1
[[2,2],[1]]
=> 2 = 3 - 1
[[2,1,1],[1]]
=> 2 = 3 - 1
[[1,1,1,1],[1]]
=> 1 = 2 - 1
[[5]]
=> 1 = 2 - 1
[[4,1]]
=> 2 = 3 - 1
[[3,2]]
=> 2 = 3 - 1
[[3,1,1]]
=> 2 = 3 - 1
[[2,2,1]]
=> 2 = 3 - 1
[[2,1,1,1]]
=> 2 = 3 - 1
[[1,1,1,1,1]]
=> 1 = 2 - 1
[[1],[1],[1],[1],[1],[1]]
=> 1 = 2 - 1
[[2],[1],[1],[1],[1]]
=> 2 = 3 - 1
[[2],[2],[1],[1]]
=> 2 = 3 - 1
Description
The number of distinct heights in a plane partition.
Matching statistic: St000318
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> 2
[[1],[1]]
=> [1,1]
=> 2
[[2]]
=> [2]
=> 2
[[1,1]]
=> [2]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> 2
[[2],[1]]
=> [2,1]
=> 3
[[1,1],[1]]
=> [2,1]
=> 3
[[3]]
=> [3]
=> 2
[[2,1]]
=> [3]
=> 2
[[1,1,1]]
=> [3]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> 3
[[2],[2]]
=> [2,2]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> 3
[[1,1],[1,1]]
=> [2,2]
=> 2
[[3],[1]]
=> [3,1]
=> 3
[[2,1],[1]]
=> [3,1]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> 3
[[4]]
=> [4]
=> 2
[[3,1]]
=> [4]
=> 2
[[2,2]]
=> [4]
=> 2
[[2,1,1]]
=> [4]
=> 2
[[1,1,1,1]]
=> [4]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 2
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 3
[[2],[2],[1]]
=> [2,2,1]
=> 3
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 3
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 3
[[3],[1],[1]]
=> [3,1,1]
=> 3
[[3],[2]]
=> [3,2]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> 3
[[2,1],[2]]
=> [3,2]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 3
[[1,1,1],[1,1]]
=> [3,2]
=> 3
[[4],[1]]
=> [4,1]
=> 3
[[3,1],[1]]
=> [4,1]
=> 3
[[2,2],[1]]
=> [4,1]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> 3
[[5]]
=> [5]
=> 2
[[4,1]]
=> [5]
=> 2
[[3,2]]
=> [5]
=> 2
[[3,1,1]]
=> [5]
=> 2
[[2,2,1]]
=> [5]
=> 2
[[2,1,1,1]]
=> [5]
=> 2
[[1,1,1,1,1]]
=> [5]
=> 2
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 2
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 3
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 3
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000159
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> 1 = 2 - 1
[[1],[1]]
=> [1,1]
=> 1 = 2 - 1
[[2]]
=> [2]
=> 1 = 2 - 1
[[1,1]]
=> [2]
=> 1 = 2 - 1
[[1],[1],[1]]
=> [1,1,1]
=> 1 = 2 - 1
[[2],[1]]
=> [2,1]
=> 2 = 3 - 1
[[1,1],[1]]
=> [2,1]
=> 2 = 3 - 1
[[3]]
=> [3]
=> 1 = 2 - 1
[[2,1]]
=> [3]
=> 1 = 2 - 1
[[1,1,1]]
=> [3]
=> 1 = 2 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> 2 = 3 - 1
[[2],[2]]
=> [2,2]
=> 1 = 2 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 2 = 3 - 1
[[1,1],[1,1]]
=> [2,2]
=> 1 = 2 - 1
[[3],[1]]
=> [3,1]
=> 2 = 3 - 1
[[2,1],[1]]
=> [3,1]
=> 2 = 3 - 1
[[1,1,1],[1]]
=> [3,1]
=> 2 = 3 - 1
[[4]]
=> [4]
=> 1 = 2 - 1
[[3,1]]
=> [4]
=> 1 = 2 - 1
[[2,2]]
=> [4]
=> 1 = 2 - 1
[[2,1,1]]
=> [4]
=> 1 = 2 - 1
[[1,1,1,1]]
=> [4]
=> 1 = 2 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 1 = 2 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 2 = 3 - 1
[[2],[2],[1]]
=> [2,2,1]
=> 2 = 3 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 2 = 3 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 2 = 3 - 1
[[3],[1],[1]]
=> [3,1,1]
=> 2 = 3 - 1
[[3],[2]]
=> [3,2]
=> 2 = 3 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 2 = 3 - 1
[[2,1],[2]]
=> [3,2]
=> 2 = 3 - 1
[[2,1],[1,1]]
=> [3,2]
=> 2 = 3 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 2 = 3 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> 2 = 3 - 1
[[4],[1]]
=> [4,1]
=> 2 = 3 - 1
[[3,1],[1]]
=> [4,1]
=> 2 = 3 - 1
[[2,2],[1]]
=> [4,1]
=> 2 = 3 - 1
[[2,1,1],[1]]
=> [4,1]
=> 2 = 3 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> 2 = 3 - 1
[[5]]
=> [5]
=> 1 = 2 - 1
[[4,1]]
=> [5]
=> 1 = 2 - 1
[[3,2]]
=> [5]
=> 1 = 2 - 1
[[3,1,1]]
=> [5]
=> 1 = 2 - 1
[[2,2,1]]
=> [5]
=> 1 = 2 - 1
[[2,1,1,1]]
=> [5]
=> 1 = 2 - 1
[[1,1,1,1,1]]
=> [5]
=> 1 = 2 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1 = 2 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 2 = 3 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 2 = 3 - 1
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: St000291
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ 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%
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 = 2 - 1
[[1],[1]]
=> [1,1]
=> 110 => 1 = 2 - 1
[[2]]
=> [2]
=> 100 => 1 = 2 - 1
[[1,1]]
=> [2]
=> 100 => 1 = 2 - 1
[[1],[1],[1]]
=> [1,1,1]
=> 1110 => 1 = 2 - 1
[[2],[1]]
=> [2,1]
=> 1010 => 2 = 3 - 1
[[1,1],[1]]
=> [2,1]
=> 1010 => 2 = 3 - 1
[[3]]
=> [3]
=> 1000 => 1 = 2 - 1
[[2,1]]
=> [3]
=> 1000 => 1 = 2 - 1
[[1,1,1]]
=> [3]
=> 1000 => 1 = 2 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 11110 => 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> 10110 => 2 = 3 - 1
[[2],[2]]
=> [2,2]
=> 1100 => 1 = 2 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 10110 => 2 = 3 - 1
[[1,1],[1,1]]
=> [2,2]
=> 1100 => 1 = 2 - 1
[[3],[1]]
=> [3,1]
=> 10010 => 2 = 3 - 1
[[2,1],[1]]
=> [3,1]
=> 10010 => 2 = 3 - 1
[[1,1,1],[1]]
=> [3,1]
=> 10010 => 2 = 3 - 1
[[4]]
=> [4]
=> 10000 => 1 = 2 - 1
[[3,1]]
=> [4]
=> 10000 => 1 = 2 - 1
[[2,2]]
=> [4]
=> 10000 => 1 = 2 - 1
[[2,1,1]]
=> [4]
=> 10000 => 1 = 2 - 1
[[1,1,1,1]]
=> [4]
=> 10000 => 1 = 2 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 111110 => 1 = 2 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 2 = 3 - 1
[[2],[2],[1]]
=> [2,2,1]
=> 11010 => 2 = 3 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 2 = 3 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 11010 => 2 = 3 - 1
[[3],[1],[1]]
=> [3,1,1]
=> 100110 => 2 = 3 - 1
[[3],[2]]
=> [3,2]
=> 10100 => 2 = 3 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 100110 => 2 = 3 - 1
[[2,1],[2]]
=> [3,2]
=> 10100 => 2 = 3 - 1
[[2,1],[1,1]]
=> [3,2]
=> 10100 => 2 = 3 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 100110 => 2 = 3 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> 10100 => 2 = 3 - 1
[[4],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[3,1],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[2,2],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[2,1,1],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[5]]
=> [5]
=> 100000 => 1 = 2 - 1
[[4,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[3,2]]
=> [5]
=> 100000 => 1 = 2 - 1
[[3,1,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[2,2,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[2,1,1,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[1,1,1,1,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1111110 => 1 = 2 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 2 = 3 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 110110 => 2 = 3 - 1
Description
The number of descents of a binary word.
Matching statistic: St000390
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ 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%
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 = 2 - 1
[[1],[1]]
=> [1,1]
=> 110 => 1 = 2 - 1
[[2]]
=> [2]
=> 100 => 1 = 2 - 1
[[1,1]]
=> [2]
=> 100 => 1 = 2 - 1
[[1],[1],[1]]
=> [1,1,1]
=> 1110 => 1 = 2 - 1
[[2],[1]]
=> [2,1]
=> 1010 => 2 = 3 - 1
[[1,1],[1]]
=> [2,1]
=> 1010 => 2 = 3 - 1
[[3]]
=> [3]
=> 1000 => 1 = 2 - 1
[[2,1]]
=> [3]
=> 1000 => 1 = 2 - 1
[[1,1,1]]
=> [3]
=> 1000 => 1 = 2 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 11110 => 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> 10110 => 2 = 3 - 1
[[2],[2]]
=> [2,2]
=> 1100 => 1 = 2 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 10110 => 2 = 3 - 1
[[1,1],[1,1]]
=> [2,2]
=> 1100 => 1 = 2 - 1
[[3],[1]]
=> [3,1]
=> 10010 => 2 = 3 - 1
[[2,1],[1]]
=> [3,1]
=> 10010 => 2 = 3 - 1
[[1,1,1],[1]]
=> [3,1]
=> 10010 => 2 = 3 - 1
[[4]]
=> [4]
=> 10000 => 1 = 2 - 1
[[3,1]]
=> [4]
=> 10000 => 1 = 2 - 1
[[2,2]]
=> [4]
=> 10000 => 1 = 2 - 1
[[2,1,1]]
=> [4]
=> 10000 => 1 = 2 - 1
[[1,1,1,1]]
=> [4]
=> 10000 => 1 = 2 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 111110 => 1 = 2 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 2 = 3 - 1
[[2],[2],[1]]
=> [2,2,1]
=> 11010 => 2 = 3 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 2 = 3 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 11010 => 2 = 3 - 1
[[3],[1],[1]]
=> [3,1,1]
=> 100110 => 2 = 3 - 1
[[3],[2]]
=> [3,2]
=> 10100 => 2 = 3 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 100110 => 2 = 3 - 1
[[2,1],[2]]
=> [3,2]
=> 10100 => 2 = 3 - 1
[[2,1],[1,1]]
=> [3,2]
=> 10100 => 2 = 3 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 100110 => 2 = 3 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> 10100 => 2 = 3 - 1
[[4],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[3,1],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[2,2],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[2,1,1],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> 100010 => 2 = 3 - 1
[[5]]
=> [5]
=> 100000 => 1 = 2 - 1
[[4,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[3,2]]
=> [5]
=> 100000 => 1 = 2 - 1
[[3,1,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[2,2,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[2,1,1,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[1,1,1,1,1]]
=> [5]
=> 100000 => 1 = 2 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1111110 => 1 = 2 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 2 = 3 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 110110 => 2 = 3 - 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000292
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> 10 => 0 = 2 - 2
[[1],[1]]
=> [1,1]
=> 110 => 0 = 2 - 2
[[2]]
=> [2]
=> 100 => 0 = 2 - 2
[[1,1]]
=> [2]
=> 100 => 0 = 2 - 2
[[1],[1],[1]]
=> [1,1,1]
=> 1110 => 0 = 2 - 2
[[2],[1]]
=> [2,1]
=> 1010 => 1 = 3 - 2
[[1,1],[1]]
=> [2,1]
=> 1010 => 1 = 3 - 2
[[3]]
=> [3]
=> 1000 => 0 = 2 - 2
[[2,1]]
=> [3]
=> 1000 => 0 = 2 - 2
[[1,1,1]]
=> [3]
=> 1000 => 0 = 2 - 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 11110 => 0 = 2 - 2
[[2],[1],[1]]
=> [2,1,1]
=> 10110 => 1 = 3 - 2
[[2],[2]]
=> [2,2]
=> 1100 => 0 = 2 - 2
[[1,1],[1],[1]]
=> [2,1,1]
=> 10110 => 1 = 3 - 2
[[1,1],[1,1]]
=> [2,2]
=> 1100 => 0 = 2 - 2
[[3],[1]]
=> [3,1]
=> 10010 => 1 = 3 - 2
[[2,1],[1]]
=> [3,1]
=> 10010 => 1 = 3 - 2
[[1,1,1],[1]]
=> [3,1]
=> 10010 => 1 = 3 - 2
[[4]]
=> [4]
=> 10000 => 0 = 2 - 2
[[3,1]]
=> [4]
=> 10000 => 0 = 2 - 2
[[2,2]]
=> [4]
=> 10000 => 0 = 2 - 2
[[2,1,1]]
=> [4]
=> 10000 => 0 = 2 - 2
[[1,1,1,1]]
=> [4]
=> 10000 => 0 = 2 - 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 111110 => 0 = 2 - 2
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 1 = 3 - 2
[[2],[2],[1]]
=> [2,2,1]
=> 11010 => 1 = 3 - 2
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => 1 = 3 - 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 11010 => 1 = 3 - 2
[[3],[1],[1]]
=> [3,1,1]
=> 100110 => 1 = 3 - 2
[[3],[2]]
=> [3,2]
=> 10100 => 1 = 3 - 2
[[2,1],[1],[1]]
=> [3,1,1]
=> 100110 => 1 = 3 - 2
[[2,1],[2]]
=> [3,2]
=> 10100 => 1 = 3 - 2
[[2,1],[1,1]]
=> [3,2]
=> 10100 => 1 = 3 - 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 100110 => 1 = 3 - 2
[[1,1,1],[1,1]]
=> [3,2]
=> 10100 => 1 = 3 - 2
[[4],[1]]
=> [4,1]
=> 100010 => 1 = 3 - 2
[[3,1],[1]]
=> [4,1]
=> 100010 => 1 = 3 - 2
[[2,2],[1]]
=> [4,1]
=> 100010 => 1 = 3 - 2
[[2,1,1],[1]]
=> [4,1]
=> 100010 => 1 = 3 - 2
[[1,1,1,1],[1]]
=> [4,1]
=> 100010 => 1 = 3 - 2
[[5]]
=> [5]
=> 100000 => 0 = 2 - 2
[[4,1]]
=> [5]
=> 100000 => 0 = 2 - 2
[[3,2]]
=> [5]
=> 100000 => 0 = 2 - 2
[[3,1,1]]
=> [5]
=> 100000 => 0 = 2 - 2
[[2,2,1]]
=> [5]
=> 100000 => 0 = 2 - 2
[[2,1,1,1]]
=> [5]
=> 100000 => 0 = 2 - 2
[[1,1,1,1,1]]
=> [5]
=> 100000 => 0 = 2 - 2
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1111110 => 0 = 2 - 2
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => 1 = 3 - 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 110110 => 1 = 3 - 2
Description
The number of ascents of a binary word.
Matching statistic: St000157
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 3
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 3
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 2
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> 2
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 3
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 3
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 3
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 3
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> 2
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> 2
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> 3
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 3
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: St000164
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000164: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000164: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 3
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 3
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> 2
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 3
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 3
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 3
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 3
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 3
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> 2
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> 2
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> 3
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 3
Description
The number of short pairs.
A short pair is a matching pair of the form (i,i+1).
Matching statistic: St000167
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [[],[]]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> 2
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 3
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 3
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 2
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[],[[[[[[]]]]]]]
=> 2
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[],[[[[],[]]]]]
=> 3
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 3
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St000069
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [[1],[]]
=> ([],1)
=> 1 = 2 - 1
[[1],[1]]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[[2]]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[[1,1]]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[2],[1]]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2 = 3 - 1
[[1,1],[1]]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2 = 3 - 1
[[3]]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[2,1]]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[1,1,1]]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 3 - 1
[[2],[2]]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 3 - 1
[[1,1],[1,1]]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 3 - 1
[[2,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 3 - 1
[[1,1,1],[1]]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 3 - 1
[[4]]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[3,1]]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[2,2]]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[2,1,1]]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[1,1,1,1]]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 3 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 3 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 3 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 3 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 3 - 1
[[3],[2]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 3 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 3 - 1
[[2,1],[2]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 3 - 1
[[2,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 3 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 3 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 3 - 1
[[4],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 3 - 1
[[3,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 3 - 1
[[2,2],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 3 - 1
[[2,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 3 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 3 - 1
[[5]]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[4,1]]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[3,2]]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[3,1,1]]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[2,2,1]]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[2,1,1,1]]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,1,1,1,1]]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2 = 3 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2 = 3 - 1
Description
The number of maximal elements of a poset.
The following 82 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000386The number of factors DDU in a Dyck path. 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. St000522The number of 1-protected nodes of a rooted tree. St000925The number of topologically connected components of a set partition. St000052The number of valleys of a Dyck path not on the x-axis. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St001083The number of boxed occurrences of 132 in a permutation. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000761The number of ascents in an integer composition. St000619The number of cyclic descents of a permutation. St000646The number of big ascents of a permutation. St000015The number of peaks of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000062The length of the longest increasing subsequence of the permutation. 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. 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. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci 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. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by τΩ1 composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000647The number of big descents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. 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. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000317The cycle descent number of a permutation. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001487The number of inner corners of a skew partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. 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). St001712The number of natural descents of a standard Young tableau.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!