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Your data matches 97 different statistics following compositions of up to 3 maps.
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Matching statistic: St000899
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St000899: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 2
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 2
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 3
[1,1,2,1] => 2
[1,1,3] => 2
[1,2,1,1] => 2
[1,2,2] => 2
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 3
[2,1,2] => 1
[2,2,1] => 2
[2,3] => 1
[3,1,1] => 2
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 4
[1,1,1,2,1] => 3
[1,1,1,3] => 3
[1,1,2,1,1] => 2
[1,1,2,2] => 2
[1,1,3,1] => 2
[1,1,4] => 2
[1,2,1,1,1] => 3
[1,2,1,2] => 1
[1,2,2,1] => 2
[1,2,3] => 1
[1,3,1,1] => 2
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 4
[2,1,1,2] => 2
[2,1,2,1] => 1
Description
The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000381
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [2] => 2
[2] => [1] => 1
[1,1,1] => [3] => 3
[1,2] => [1,1] => 1
[2,1] => [1,1] => 1
[3] => [1] => 1
[1,1,1,1] => [4] => 4
[1,1,2] => [2,1] => 2
[1,2,1] => [1,1,1] => 1
[1,3] => [1,1] => 1
[2,1,1] => [1,2] => 2
[2,2] => [2] => 2
[3,1] => [1,1] => 1
[4] => [1] => 1
[1,1,1,1,1] => [5] => 5
[1,1,1,2] => [3,1] => 3
[1,1,2,1] => [2,1,1] => 2
[1,1,3] => [2,1] => 2
[1,2,1,1] => [1,1,2] => 2
[1,2,2] => [1,2] => 2
[1,3,1] => [1,1,1] => 1
[1,4] => [1,1] => 1
[2,1,1,1] => [1,3] => 3
[2,1,2] => [1,1,1] => 1
[2,2,1] => [2,1] => 2
[2,3] => [1,1] => 1
[3,1,1] => [1,2] => 2
[3,2] => [1,1] => 1
[4,1] => [1,1] => 1
[5] => [1] => 1
[1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [4,1] => 4
[1,1,1,2,1] => [3,1,1] => 3
[1,1,1,3] => [3,1] => 3
[1,1,2,1,1] => [2,1,2] => 2
[1,1,2,2] => [2,2] => 2
[1,1,3,1] => [2,1,1] => 2
[1,1,4] => [2,1] => 2
[1,2,1,1,1] => [1,1,3] => 3
[1,2,1,2] => [1,1,1,1] => 1
[1,2,2,1] => [1,2,1] => 2
[1,2,3] => [1,1,1] => 1
[1,3,1,1] => [1,1,2] => 2
[1,3,2] => [1,1,1] => 1
[1,4,1] => [1,1,1] => 1
[1,5] => [1,1] => 1
[2,1,1,1,1] => [1,4] => 4
[2,1,1,2] => [1,2,1] => 2
[2,1,2,1] => [1,1,1,1] => 1
Description
The largest part of an integer composition.
Matching statistic: St000147
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,1] => [2] => [2]
=> 2
[2] => [1] => [1]
=> 1
[1,1,1] => [3] => [3]
=> 3
[1,2] => [1,1] => [1,1]
=> 1
[2,1] => [1,1] => [1,1]
=> 1
[3] => [1] => [1]
=> 1
[1,1,1,1] => [4] => [4]
=> 4
[1,1,2] => [2,1] => [2,1]
=> 2
[1,2,1] => [1,1,1] => [1,1,1]
=> 1
[1,3] => [1,1] => [1,1]
=> 1
[2,1,1] => [1,2] => [2,1]
=> 2
[2,2] => [2] => [2]
=> 2
[3,1] => [1,1] => [1,1]
=> 1
[4] => [1] => [1]
=> 1
[1,1,1,1,1] => [5] => [5]
=> 5
[1,1,1,2] => [3,1] => [3,1]
=> 3
[1,1,2,1] => [2,1,1] => [2,1,1]
=> 2
[1,1,3] => [2,1] => [2,1]
=> 2
[1,2,1,1] => [1,1,2] => [2,1,1]
=> 2
[1,2,2] => [1,2] => [2,1]
=> 2
[1,3,1] => [1,1,1] => [1,1,1]
=> 1
[1,4] => [1,1] => [1,1]
=> 1
[2,1,1,1] => [1,3] => [3,1]
=> 3
[2,1,2] => [1,1,1] => [1,1,1]
=> 1
[2,2,1] => [2,1] => [2,1]
=> 2
[2,3] => [1,1] => [1,1]
=> 1
[3,1,1] => [1,2] => [2,1]
=> 2
[3,2] => [1,1] => [1,1]
=> 1
[4,1] => [1,1] => [1,1]
=> 1
[5] => [1] => [1]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> 6
[1,1,1,1,2] => [4,1] => [4,1]
=> 4
[1,1,1,2,1] => [3,1,1] => [3,1,1]
=> 3
[1,1,1,3] => [3,1] => [3,1]
=> 3
[1,1,2,1,1] => [2,1,2] => [2,2,1]
=> 2
[1,1,2,2] => [2,2] => [2,2]
=> 2
[1,1,3,1] => [2,1,1] => [2,1,1]
=> 2
[1,1,4] => [2,1] => [2,1]
=> 2
[1,2,1,1,1] => [1,1,3] => [3,1,1]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,1,1,1]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1]
=> 2
[1,2,3] => [1,1,1] => [1,1,1]
=> 1
[1,3,1,1] => [1,1,2] => [2,1,1]
=> 2
[1,3,2] => [1,1,1] => [1,1,1]
=> 1
[1,4,1] => [1,1,1] => [1,1,1]
=> 1
[1,5] => [1,1] => [1,1]
=> 1
[2,1,1,1,1] => [1,4] => [4,1]
=> 4
[2,1,1,2] => [1,2,1] => [2,1,1]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 1
[1,1] => [2] => [2]
=> [1,1]
=> 2
[2] => [1] => [1]
=> [1]
=> 1
[1,1,1] => [3] => [3]
=> [1,1,1]
=> 3
[1,2] => [1,1] => [1,1]
=> [2]
=> 1
[2,1] => [1,1] => [1,1]
=> [2]
=> 1
[3] => [1] => [1]
=> [1]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,1,1,1]
=> 4
[1,1,2] => [2,1] => [2,1]
=> [2,1]
=> 2
[1,2,1] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,3] => [1,1] => [1,1]
=> [2]
=> 1
[2,1,1] => [1,2] => [2,1]
=> [2,1]
=> 2
[2,2] => [2] => [2]
=> [1,1]
=> 2
[3,1] => [1,1] => [1,1]
=> [2]
=> 1
[4] => [1] => [1]
=> [1]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,1,1,1,1]
=> 5
[1,1,1,2] => [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,1,2,1] => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[1,1,3] => [2,1] => [2,1]
=> [2,1]
=> 2
[1,2,1,1] => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
[1,2,2] => [1,2] => [2,1]
=> [2,1]
=> 2
[1,3,1] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,4] => [1,1] => [1,1]
=> [2]
=> 1
[2,1,1,1] => [1,3] => [3,1]
=> [2,1,1]
=> 3
[2,1,2] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[2,2,1] => [2,1] => [2,1]
=> [2,1]
=> 2
[2,3] => [1,1] => [1,1]
=> [2]
=> 1
[3,1,1] => [1,2] => [2,1]
=> [2,1]
=> 2
[3,2] => [1,1] => [1,1]
=> [2]
=> 1
[4,1] => [1,1] => [1,1]
=> [2]
=> 1
[5] => [1] => [1]
=> [1]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [4,1] => [4,1]
=> [2,1,1,1]
=> 4
[1,1,1,2,1] => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,1,3] => [3,1] => [3,1]
=> [2,1,1]
=> 3
[1,1,2,1,1] => [2,1,2] => [2,2,1]
=> [3,2]
=> 2
[1,1,2,2] => [2,2] => [2,2]
=> [2,2]
=> 2
[1,1,3,1] => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[1,1,4] => [2,1] => [2,1]
=> [2,1]
=> 2
[1,2,1,1,1] => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[1,2,3] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,3,1,1] => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
[1,3,2] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,4,1] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,5] => [1,1] => [1,1]
=> [2]
=> 1
[2,1,1,1,1] => [1,4] => [4,1]
=> [2,1,1,1]
=> 4
[2,1,1,2] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
Description
The length of the partition.
Matching statistic: St000734
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [[1]]
=> 1
[1,1] => [2] => [2]
=> [[1,2]]
=> 2
[2] => [1] => [1]
=> [[1]]
=> 1
[1,1,1] => [3] => [3]
=> [[1,2,3]]
=> 3
[1,2] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[2,1] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[3] => [1] => [1]
=> [[1]]
=> 1
[1,1,1,1] => [4] => [4]
=> [[1,2,3,4]]
=> 4
[1,1,2] => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[1,2,1] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[2,1,1] => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[2,2] => [2] => [2]
=> [[1,2]]
=> 2
[3,1] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[4] => [1] => [1]
=> [[1]]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [[1,2,3,4,5]]
=> 5
[1,1,1,2] => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,2,1] => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,3] => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[1,2,1,1] => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,2,2] => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[1,3,1] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,4] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[2,1,1,1] => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
[2,1,2] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[2,2,1] => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[2,3] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[3,1,1] => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[3,2] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[4,1] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[5] => [1] => [1]
=> [[1]]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [[1,2,3,4,5,6]]
=> 6
[1,1,1,1,2] => [4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,1,1,2,1] => [3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,1,1,3] => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[1,1,2,1,1] => [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,1,2,2] => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,3,1] => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,4] => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[1,2,1,1,1] => [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,2,3] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3,1,1] => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,4,1] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,5] => [1,1] => [1,1]
=> [[1],[2]]
=> 1
[2,1,1,1,1] => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[2,1,1,2] => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000983
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 1 => 1
[1,1] => [2] => 10 => 01 => 2
[2] => [1] => 1 => 1 => 1
[1,1,1] => [3] => 100 => 101 => 3
[1,2] => [1,1] => 11 => 11 => 1
[2,1] => [1,1] => 11 => 11 => 1
[3] => [1] => 1 => 1 => 1
[1,1,1,1] => [4] => 1000 => 0101 => 4
[1,1,2] => [2,1] => 101 => 001 => 2
[1,2,1] => [1,1,1] => 111 => 111 => 1
[1,3] => [1,1] => 11 => 11 => 1
[2,1,1] => [1,2] => 110 => 011 => 2
[2,2] => [2] => 10 => 01 => 2
[3,1] => [1,1] => 11 => 11 => 1
[4] => [1] => 1 => 1 => 1
[1,1,1,1,1] => [5] => 10000 => 10101 => 5
[1,1,1,2] => [3,1] => 1001 => 1101 => 3
[1,1,2,1] => [2,1,1] => 1011 => 0001 => 2
[1,1,3] => [2,1] => 101 => 001 => 2
[1,2,1,1] => [1,1,2] => 1110 => 0111 => 2
[1,2,2] => [1,2] => 110 => 011 => 2
[1,3,1] => [1,1,1] => 111 => 111 => 1
[1,4] => [1,1] => 11 => 11 => 1
[2,1,1,1] => [1,3] => 1100 => 1011 => 3
[2,1,2] => [1,1,1] => 111 => 111 => 1
[2,2,1] => [2,1] => 101 => 001 => 2
[2,3] => [1,1] => 11 => 11 => 1
[3,1,1] => [1,2] => 110 => 011 => 2
[3,2] => [1,1] => 11 => 11 => 1
[4,1] => [1,1] => 11 => 11 => 1
[5] => [1] => 1 => 1 => 1
[1,1,1,1,1,1] => [6] => 100000 => 010101 => 6
[1,1,1,1,2] => [4,1] => 10001 => 00101 => 4
[1,1,1,2,1] => [3,1,1] => 10011 => 11101 => 3
[1,1,1,3] => [3,1] => 1001 => 1101 => 3
[1,1,2,1,1] => [2,1,2] => 10110 => 10001 => 2
[1,1,2,2] => [2,2] => 1010 => 1001 => 2
[1,1,3,1] => [2,1,1] => 1011 => 0001 => 2
[1,1,4] => [2,1] => 101 => 001 => 2
[1,2,1,1,1] => [1,1,3] => 11100 => 10111 => 3
[1,2,1,2] => [1,1,1,1] => 1111 => 1111 => 1
[1,2,2,1] => [1,2,1] => 1101 => 0011 => 2
[1,2,3] => [1,1,1] => 111 => 111 => 1
[1,3,1,1] => [1,1,2] => 1110 => 0111 => 2
[1,3,2] => [1,1,1] => 111 => 111 => 1
[1,4,1] => [1,1,1] => 111 => 111 => 1
[1,5] => [1,1] => 11 => 11 => 1
[2,1,1,1,1] => [1,4] => 11000 => 01011 => 4
[2,1,1,2] => [1,2,1] => 1101 => 0011 => 2
[2,1,2,1] => [1,1,1,1] => 1111 => 1111 => 1
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 0 => 0 = 1 - 1
[1,1] => [2] => 10 => 01 => 1 = 2 - 1
[2] => [1] => 1 => 0 => 0 = 1 - 1
[1,1,1] => [3] => 100 => 011 => 2 = 3 - 1
[1,2] => [1,1] => 11 => 00 => 0 = 1 - 1
[2,1] => [1,1] => 11 => 00 => 0 = 1 - 1
[3] => [1] => 1 => 0 => 0 = 1 - 1
[1,1,1,1] => [4] => 1000 => 0111 => 3 = 4 - 1
[1,1,2] => [2,1] => 101 => 010 => 1 = 2 - 1
[1,2,1] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,3] => [1,1] => 11 => 00 => 0 = 1 - 1
[2,1,1] => [1,2] => 110 => 001 => 1 = 2 - 1
[2,2] => [2] => 10 => 01 => 1 = 2 - 1
[3,1] => [1,1] => 11 => 00 => 0 = 1 - 1
[4] => [1] => 1 => 0 => 0 = 1 - 1
[1,1,1,1,1] => [5] => 10000 => 01111 => 4 = 5 - 1
[1,1,1,2] => [3,1] => 1001 => 0110 => 2 = 3 - 1
[1,1,2,1] => [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[1,1,3] => [2,1] => 101 => 010 => 1 = 2 - 1
[1,2,1,1] => [1,1,2] => 1110 => 0001 => 1 = 2 - 1
[1,2,2] => [1,2] => 110 => 001 => 1 = 2 - 1
[1,3,1] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,4] => [1,1] => 11 => 00 => 0 = 1 - 1
[2,1,1,1] => [1,3] => 1100 => 0011 => 2 = 3 - 1
[2,1,2] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[2,2,1] => [2,1] => 101 => 010 => 1 = 2 - 1
[2,3] => [1,1] => 11 => 00 => 0 = 1 - 1
[3,1,1] => [1,2] => 110 => 001 => 1 = 2 - 1
[3,2] => [1,1] => 11 => 00 => 0 = 1 - 1
[4,1] => [1,1] => 11 => 00 => 0 = 1 - 1
[5] => [1] => 1 => 0 => 0 = 1 - 1
[1,1,1,1,1,1] => [6] => 100000 => 011111 => 5 = 6 - 1
[1,1,1,1,2] => [4,1] => 10001 => 01110 => 3 = 4 - 1
[1,1,1,2,1] => [3,1,1] => 10011 => 01100 => 2 = 3 - 1
[1,1,1,3] => [3,1] => 1001 => 0110 => 2 = 3 - 1
[1,1,2,1,1] => [2,1,2] => 10110 => 01001 => 1 = 2 - 1
[1,1,2,2] => [2,2] => 1010 => 0101 => 1 = 2 - 1
[1,1,3,1] => [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[1,1,4] => [2,1] => 101 => 010 => 1 = 2 - 1
[1,2,1,1,1] => [1,1,3] => 11100 => 00011 => 2 = 3 - 1
[1,2,1,2] => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
[1,2,2,1] => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
[1,2,3] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,3,1,1] => [1,1,2] => 1110 => 0001 => 1 = 2 - 1
[1,3,2] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,4,1] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,5] => [1,1] => 11 => 00 => 0 = 1 - 1
[2,1,1,1,1] => [1,4] => 11000 => 00111 => 3 = 4 - 1
[2,1,1,2] => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
[2,1,2,1] => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000013
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000013: 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
[1,1] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 2
[2] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,2] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[3] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,3] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[4] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,4] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,3] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[4,1] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[5] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,5] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 4
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000676
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 100%●distinct values known / distinct values provided: 89%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 100%●distinct values known / distinct values provided: 89%
Values
[1] => [1] => [1]
=> [1,0]
=> 1
[1,1] => [2] => [2]
=> [1,0,1,0]
=> 2
[2] => [1] => [1]
=> [1,0]
=> 1
[1,1,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[2,1] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[3] => [1] => [1]
=> [1,0]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[2,1,1] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,2] => [2] => [2]
=> [1,0,1,0]
=> 2
[3,1] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[4] => [1] => [1]
=> [1,0]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,2,1] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,3] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,1,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,2] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,1] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[2,1,1,1] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,2] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,2,1] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[3,1,1] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[4,1] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[5] => [1] => [1]
=> [1,0]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,2,1] => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,1,3] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,2,1,1] => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,2,2] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,3,1] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,4] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,1,1,1] => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,2,3] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,1,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,1] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[2,1,1,1,1] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,1,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000684
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 99%●distinct values known / distinct values provided: 89%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 99%●distinct values known / distinct values provided: 89%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1,1] => [1,0,1,0]
=> 2
[2] => [1] => [1] => [1,0]
=> 1
[1,1,1] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,1] => [2] => [1,1,0,0]
=> 1
[2,1] => [1,1] => [2] => [1,1,0,0]
=> 1
[3] => [1] => [1] => [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,2,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,3] => [1,1] => [2] => [1,1,0,0]
=> 1
[2,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,2] => [2] => [1,1] => [1,0,1,0]
=> 2
[3,1] => [1,1] => [2] => [1,1,0,0]
=> 1
[4] => [1] => [1] => [1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,1,2,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,3] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,2,1,1] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,2,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,3,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,4] => [1,1] => [2] => [1,1,0,0]
=> 1
[2,1,1,1] => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,1,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[2,2,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,3] => [1,1] => [2] => [1,1,0,0]
=> 1
[3,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2] => [1,1] => [2] => [1,1,0,0]
=> 1
[4,1] => [1,1] => [2] => [1,1,0,0]
=> 1
[5] => [1] => [1] => [1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,2,1] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,3] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,1,2,1,1] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,2,2] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,4] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,2,1,1,1] => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,3] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,3,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,4,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,5] => [1,1] => [2] => [1,1,0,0]
=> 1
[2,1,1,1,1] => [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,1,2] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,9}
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,9}
[1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,9}
Description
The global dimension of the LNakayama algebra associated to a Dyck path.
An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$.
The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$.
One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0].
Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:
* For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.
* For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
The following 87 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000025The number of initial rises of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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:
St001809The index of the step at the first peak of maximal height in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001062The maximal size of a block of a set partition. St000662The staircase size of the code of a permutation. St000306The bounce count of a Dyck path. St000141The maximum drop size of a permutation. St000308The height of the tree associated to a permutation. St000209Maximum difference of elements in cycles. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001652The length of a longest interval of consecutive numbers. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000442The maximal area to the right of an up step of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000094The depth of an ordered tree. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000028The number of stack-sorts needed to sort a permutation. St001589The nesting number of a perfect matching. St001330The hat guessing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St001933The largest multiplicity of a part in an integer partition. St000922The minimal number such that all substrings of this length are unique. St000120The number of left tunnels of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000678The number of up steps after the last double rise of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001116The game chromatic number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000731The number of double exceedences of a permutation. St001875The number of simple modules with projective dimension at most 1. St001642The Prague dimension of a graph. St000822The Hadwiger number of the graph. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001118The acyclic chromatic index of a graph.
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