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Your data matches 216 different statistics following compositions of up to 3 maps.
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Matching statistic: St000381
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
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
[1,1,1] => 1
[1,2] => 2
[2,1] => 2
[3] => 3
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 3
[2,1,1] => 2
[2,2] => 2
[3,1] => 3
[4] => 4
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 2
[1,1,3] => 3
[1,2,1,1] => 2
[1,2,2] => 2
[1,3,1] => 3
[1,4] => 4
[2,1,1,1] => 2
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 3
[3,1,1] => 3
[3,2] => 3
[4,1] => 4
[5] => 5
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 2
[1,1,1,3] => 3
[1,1,2,1,1] => 2
[1,1,2,2] => 2
[1,1,3,1] => 3
[1,1,4] => 4
[1,2,1,1,1] => 2
[1,2,1,2] => 2
[1,2,2,1] => 2
[1,2,3] => 3
[1,3,1,1] => 3
[1,3,2] => 3
[1,4,1] => 4
[1,5] => 5
[2,1,1,1,1] => 2
[2,1,1,2] => 2
[2,1,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
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%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,1] => [1,1]
=> 1
[2] => [2]
=> 2
[1,1,1] => [1,1,1]
=> 1
[1,2] => [2,1]
=> 2
[2,1] => [2,1]
=> 2
[3] => [3]
=> 3
[1,1,1,1] => [1,1,1,1]
=> 1
[1,1,2] => [2,1,1]
=> 2
[1,2,1] => [2,1,1]
=> 2
[1,3] => [3,1]
=> 3
[2,1,1] => [2,1,1]
=> 2
[2,2] => [2,2]
=> 2
[3,1] => [3,1]
=> 3
[4] => [4]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> 1
[1,1,1,2] => [2,1,1,1]
=> 2
[1,1,2,1] => [2,1,1,1]
=> 2
[1,1,3] => [3,1,1]
=> 3
[1,2,1,1] => [2,1,1,1]
=> 2
[1,2,2] => [2,2,1]
=> 2
[1,3,1] => [3,1,1]
=> 3
[1,4] => [4,1]
=> 4
[2,1,1,1] => [2,1,1,1]
=> 2
[2,1,2] => [2,2,1]
=> 2
[2,2,1] => [2,2,1]
=> 2
[2,3] => [3,2]
=> 3
[3,1,1] => [3,1,1]
=> 3
[3,2] => [3,2]
=> 3
[4,1] => [4,1]
=> 4
[5] => [5]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> 2
[1,1,1,3] => [3,1,1,1]
=> 3
[1,1,2,1,1] => [2,1,1,1,1]
=> 2
[1,1,2,2] => [2,2,1,1]
=> 2
[1,1,3,1] => [3,1,1,1]
=> 3
[1,1,4] => [4,1,1]
=> 4
[1,2,1,1,1] => [2,1,1,1,1]
=> 2
[1,2,1,2] => [2,2,1,1]
=> 2
[1,2,2,1] => [2,2,1,1]
=> 2
[1,2,3] => [3,2,1]
=> 3
[1,3,1,1] => [3,1,1,1]
=> 3
[1,3,2] => [3,2,1]
=> 3
[1,4,1] => [4,1,1]
=> 4
[1,5] => [5,1]
=> 5
[2,1,1,1,1] => [2,1,1,1,1]
=> 2
[2,1,1,2] => [2,2,1,1]
=> 2
[2,1,2,1] => [2,2,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000982
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1
[1,1] => 11 => 2
[2] => 10 => 1
[1,1,1] => 111 => 3
[1,2] => 110 => 2
[2,1] => 101 => 1
[3] => 100 => 2
[1,1,1,1] => 1111 => 4
[1,1,2] => 1110 => 3
[1,2,1] => 1101 => 2
[1,3] => 1100 => 2
[2,1,1] => 1011 => 2
[2,2] => 1010 => 1
[3,1] => 1001 => 2
[4] => 1000 => 3
[1,1,1,1,1] => 11111 => 5
[1,1,1,2] => 11110 => 4
[1,1,2,1] => 11101 => 3
[1,1,3] => 11100 => 3
[1,2,1,1] => 11011 => 2
[1,2,2] => 11010 => 2
[1,3,1] => 11001 => 2
[1,4] => 11000 => 3
[2,1,1,1] => 10111 => 3
[2,1,2] => 10110 => 2
[2,2,1] => 10101 => 1
[2,3] => 10100 => 2
[3,1,1] => 10011 => 2
[3,2] => 10010 => 2
[4,1] => 10001 => 3
[5] => 10000 => 4
[1,1,1,1,1,1] => 111111 => 6
[1,1,1,1,2] => 111110 => 5
[1,1,1,2,1] => 111101 => 4
[1,1,1,3] => 111100 => 4
[1,1,2,1,1] => 111011 => 3
[1,1,2,2] => 111010 => 3
[1,1,3,1] => 111001 => 3
[1,1,4] => 111000 => 3
[1,2,1,1,1] => 110111 => 3
[1,2,1,2] => 110110 => 2
[1,2,2,1] => 110101 => 2
[1,2,3] => 110100 => 2
[1,3,1,1] => 110011 => 2
[1,3,2] => 110010 => 2
[1,4,1] => 110001 => 3
[1,5] => 110000 => 4
[2,1,1,1,1] => 101111 => 4
[2,1,1,2] => 101110 => 3
[2,1,2,1] => 101101 => 2
Description
The length of the longest constant subword.
Matching statistic: St000983
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1
[1,1] => 11 => 1
[2] => 10 => 2
[1,1,1] => 111 => 1
[1,2] => 110 => 2
[2,1] => 101 => 3
[3] => 100 => 2
[1,1,1,1] => 1111 => 1
[1,1,2] => 1110 => 2
[1,2,1] => 1101 => 3
[1,3] => 1100 => 2
[2,1,1] => 1011 => 3
[2,2] => 1010 => 4
[3,1] => 1001 => 2
[4] => 1000 => 2
[1,1,1,1,1] => 11111 => 1
[1,1,1,2] => 11110 => 2
[1,1,2,1] => 11101 => 3
[1,1,3] => 11100 => 2
[1,2,1,1] => 11011 => 3
[1,2,2] => 11010 => 4
[1,3,1] => 11001 => 2
[1,4] => 11000 => 2
[2,1,1,1] => 10111 => 3
[2,1,2] => 10110 => 3
[2,2,1] => 10101 => 5
[2,3] => 10100 => 4
[3,1,1] => 10011 => 2
[3,2] => 10010 => 3
[4,1] => 10001 => 2
[5] => 10000 => 2
[1,1,1,1,1,1] => 111111 => 1
[1,1,1,1,2] => 111110 => 2
[1,1,1,2,1] => 111101 => 3
[1,1,1,3] => 111100 => 2
[1,1,2,1,1] => 111011 => 3
[1,1,2,2] => 111010 => 4
[1,1,3,1] => 111001 => 2
[1,1,4] => 111000 => 2
[1,2,1,1,1] => 110111 => 3
[1,2,1,2] => 110110 => 3
[1,2,2,1] => 110101 => 5
[1,2,3] => 110100 => 4
[1,3,1,1] => 110011 => 2
[1,3,2] => 110010 => 3
[1,4,1] => 110001 => 2
[1,5] => 110000 => 2
[2,1,1,1,1] => 101111 => 3
[2,1,1,2] => 101110 => 3
[2,1,2,1] => 101101 => 3
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: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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] => [1,1]
=> [2]
=> 1
[2] => [2]
=> [1,1]
=> 2
[1,1,1] => [1,1,1]
=> [3]
=> 1
[1,2] => [2,1]
=> [2,1]
=> 2
[2,1] => [2,1]
=> [2,1]
=> 2
[3] => [3]
=> [1,1,1]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,1,2] => [2,1,1]
=> [3,1]
=> 2
[1,2,1] => [2,1,1]
=> [3,1]
=> 2
[1,3] => [3,1]
=> [2,1,1]
=> 3
[2,1,1] => [2,1,1]
=> [3,1]
=> 2
[2,2] => [2,2]
=> [2,2]
=> 2
[3,1] => [3,1]
=> [2,1,1]
=> 3
[4] => [4]
=> [1,1,1,1]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[1,2,2] => [2,2,1]
=> [3,2]
=> 2
[1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,4] => [4,1]
=> [2,1,1,1]
=> 4
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
[2,1,2] => [2,2,1]
=> [3,2]
=> 2
[2,2,1] => [2,2,1]
=> [3,2]
=> 2
[2,3] => [3,2]
=> [2,2,1]
=> 3
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 3
[3,2] => [3,2]
=> [2,2,1]
=> 3
[4,1] => [4,1]
=> [2,1,1,1]
=> 4
[5] => [5]
=> [1,1,1,1,1]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> 2
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> 3
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> 2
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> 2
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> 3
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 4
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 2
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> 2
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> 2
[1,2,3] => [3,2,1]
=> [3,2,1]
=> 3
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 3
[1,3,2] => [3,2,1]
=> [3,2,1]
=> 3
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 4
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 5
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 2
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 2
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 2
Description
The length of the partition.
Matching statistic: St000734
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%
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] => [1,1]
=> [[1],[2]]
=> 1
[2] => [2]
=> [[1,2]]
=> 2
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,2] => [2,1]
=> [[1,2],[3]]
=> 2
[2,1] => [2,1]
=> [[1,2],[3]]
=> 2
[3] => [3]
=> [[1,2,3]]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
[2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
[4] => [4]
=> [[1,2,3,4]]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
[4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
[5] => [5]
=> [[1,2,3,4,5]]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,1,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
[1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
[1,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000392
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
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%
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 => 0 => 0 = 1 - 1
[1,1] => 11 => 00 => 0 = 1 - 1
[2] => 10 => 01 => 1 = 2 - 1
[1,1,1] => 111 => 000 => 0 = 1 - 1
[1,2] => 110 => 001 => 1 = 2 - 1
[2,1] => 101 => 010 => 1 = 2 - 1
[3] => 100 => 011 => 2 = 3 - 1
[1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
[1,1,2] => 1110 => 0001 => 1 = 2 - 1
[1,2,1] => 1101 => 0010 => 1 = 2 - 1
[1,3] => 1100 => 0011 => 2 = 3 - 1
[2,1,1] => 1011 => 0100 => 1 = 2 - 1
[2,2] => 1010 => 0101 => 1 = 2 - 1
[3,1] => 1001 => 0110 => 2 = 3 - 1
[4] => 1000 => 0111 => 3 = 4 - 1
[1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
[1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
[1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
[1,1,3] => 11100 => 00011 => 2 = 3 - 1
[1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
[1,2,2] => 11010 => 00101 => 1 = 2 - 1
[1,3,1] => 11001 => 00110 => 2 = 3 - 1
[1,4] => 11000 => 00111 => 3 = 4 - 1
[2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[2,1,2] => 10110 => 01001 => 1 = 2 - 1
[2,2,1] => 10101 => 01010 => 1 = 2 - 1
[2,3] => 10100 => 01011 => 2 = 3 - 1
[3,1,1] => 10011 => 01100 => 2 = 3 - 1
[3,2] => 10010 => 01101 => 2 = 3 - 1
[4,1] => 10001 => 01110 => 3 = 4 - 1
[5] => 10000 => 01111 => 4 = 5 - 1
[1,1,1,1,1,1] => 111111 => 000000 => 0 = 1 - 1
[1,1,1,1,2] => 111110 => 000001 => 1 = 2 - 1
[1,1,1,2,1] => 111101 => 000010 => 1 = 2 - 1
[1,1,1,3] => 111100 => 000011 => 2 = 3 - 1
[1,1,2,1,1] => 111011 => 000100 => 1 = 2 - 1
[1,1,2,2] => 111010 => 000101 => 1 = 2 - 1
[1,1,3,1] => 111001 => 000110 => 2 = 3 - 1
[1,1,4] => 111000 => 000111 => 3 = 4 - 1
[1,2,1,1,1] => 110111 => 001000 => 1 = 2 - 1
[1,2,1,2] => 110110 => 001001 => 1 = 2 - 1
[1,2,2,1] => 110101 => 001010 => 1 = 2 - 1
[1,2,3] => 110100 => 001011 => 2 = 3 - 1
[1,3,1,1] => 110011 => 001100 => 2 = 3 - 1
[1,3,2] => 110010 => 001101 => 2 = 3 - 1
[1,4,1] => 110001 => 001110 => 3 = 4 - 1
[1,5] => 110000 => 001111 => 4 = 5 - 1
[2,1,1,1,1] => 101111 => 010000 => 1 = 2 - 1
[2,1,1,2] => 101110 => 010001 => 1 = 2 - 1
[2,1,2,1] => 101101 => 010010 => 1 = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000013
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 1
[1,1] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
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: St000141
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ 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] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[2] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[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]
=> [2,3,4,5,6,7,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 2
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 2
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 4
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 2
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 4
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 5
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 2
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 10 => 1
[1,1] => [1,1]
=> [2]
=> 100 => 1
[2] => [2]
=> [1,1]
=> 110 => 2
[1,1,1] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2] => [2,1]
=> [2,1]
=> 1010 => 2
[2,1] => [2,1]
=> [2,1]
=> 1010 => 2
[3] => [3]
=> [1,1,1]
=> 1110 => 3
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 10000 => 1
[1,1,2] => [2,1,1]
=> [3,1]
=> 10010 => 2
[1,2,1] => [2,1,1]
=> [3,1]
=> 10010 => 2
[1,3] => [3,1]
=> [2,1,1]
=> 10110 => 3
[2,1,1] => [2,1,1]
=> [3,1]
=> 10010 => 2
[2,2] => [2,2]
=> [2,2]
=> 1100 => 2
[3,1] => [3,1]
=> [2,1,1]
=> 10110 => 3
[4] => [4]
=> [1,1,1,1]
=> 11110 => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 100000 => 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 100010 => 2
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 2
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 100110 => 3
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 2
[1,2,2] => [2,2,1]
=> [3,2]
=> 10100 => 2
[1,3,1] => [3,1,1]
=> [3,1,1]
=> 100110 => 3
[1,4] => [4,1]
=> [2,1,1,1]
=> 101110 => 4
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 2
[2,1,2] => [2,2,1]
=> [3,2]
=> 10100 => 2
[2,2,1] => [2,2,1]
=> [3,2]
=> 10100 => 2
[2,3] => [3,2]
=> [2,2,1]
=> 11010 => 3
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 100110 => 3
[3,2] => [3,2]
=> [2,2,1]
=> 11010 => 3
[4,1] => [4,1]
=> [2,1,1,1]
=> 101110 => 4
[5] => [5]
=> [1,1,1,1,1]
=> 111110 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 2
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 3
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 2
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 2
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 3
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 1001110 => 4
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 2
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 2
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> 100100 => 2
[1,2,3] => [3,2,1]
=> [3,2,1]
=> 101010 => 3
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 3
[1,3,2] => [3,2,1]
=> [3,2,1]
=> 101010 => 3
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 1001110 => 4
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 1011110 => 5
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 2
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 2
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 100100 => 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 206 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000326The position of the first one in a binary word after appending a 1 at the end. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000808The number of up steps of the associated bargraph. St000054The first entry of the permutation. St000157The number of descents of a standard tableau. St000225Difference between largest and smallest parts in a partition. St001372The length of a longest cyclic run of ones of a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000691The number of changes of a binary word. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000439The position of the first down step of a Dyck path. 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. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000025The number of initial rises of a Dyck path. 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. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000653The last descent of 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. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001497The position of the largest weak excedence of a permutation. St000738The first entry in the last row of a standard tableau. St000011The number of touch points (or returns) of a Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000702The number of weak deficiencies of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000306The bounce count of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000093The cardinality of a maximal independent set of vertices of a graph. St001645The pebbling number of a connected graph. St001058The breadth of the ordered tree. St000097The order of the largest clique of the graph. St001060The distinguishing index of a graph. St000443The number of long tunnels of a Dyck path. St000991The number of right-to-left minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001180Number of indecomposable injective modules with projective dimension at most 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000083The number of left oriented leafs of a binary tree except the first one. St000383The last part of an integer composition. St000840The number of closers smaller than the largest opener in a perfect matching. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000662The staircase size of the code of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000098The chromatic number of a graph. St001644The dimension of a graph. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000527The width of the poset. St000823The number of unsplittable factors of the set partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000075The orbit size of a standard tableau under promotion. St000686The finitistic dominant dimension of 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:
St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000053The number of valleys of the Dyck path. St000171The degree of the graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001826The maximal number of leaves on a vertex of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000730The maximal arc length of a set partition. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St001120The length of a longest path in a graph. St001963The tree-depth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000264The girth of a graph, which is not a tree. St000028The number of stack-sorts needed to sort a permutation. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. 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 $eA$. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St000308The height of the tree associated to a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000651The maximal size of a rise in a permutation. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000245The number of ascents of a permutation. St001652The length of a longest interval of consecutive numbers. St001717The largest size of an interval in a poset. St000746The number of pairs with odd minimum in a perfect matching. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000470The number of runs in a permutation. St001330The hat guessing number of a graph. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000542The number of left-to-right-minima of a permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. 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. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. 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 $A$. St000015The number of peaks of a Dyck path. St000297The number of leading ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000021The number of descents of a permutation. St000080The rank of the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000989The number of final rises of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001117The game chromatic index of a graph. 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 $eA$. 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 $\tau \Omega^1$ 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. St000061The number of nodes on the left branch of a binary tree. St001118The acyclic chromatic index of a graph. St000652The maximal difference between successive positions of a permutation. St001654The monophonic hull number of a graph. St001875The number of simple modules with projective dimension at most 1. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001589The nesting number of a perfect matching. St000155The number of exceedances (also excedences) of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. 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 $eA$. St001590The crossing number of a perfect matching. St000317The cycle descent number of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000455The second largest eigenvalue of a graph if it is integral.
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