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Your data matches 191 different statistics following compositions of up to 3 maps.
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Matching statistic: St000225
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St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0 = 1 - 1
[2]
=> 0 = 1 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 0 = 1 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 0 = 1 - 1
[4]
=> 0 = 1 - 1
[3,1]
=> 2 = 3 - 1
[2,2]
=> 0 = 1 - 1
[2,1,1]
=> 1 = 2 - 1
[1,1,1,1]
=> 0 = 1 - 1
[5]
=> 0 = 1 - 1
[4,1]
=> 3 = 4 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 2 = 3 - 1
[2,2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> 0 = 1 - 1
[6]
=> 0 = 1 - 1
[5,1]
=> 4 = 5 - 1
[4,2]
=> 2 = 3 - 1
[4,1,1]
=> 3 = 4 - 1
[3,3]
=> 0 = 1 - 1
[3,2,1]
=> 2 = 3 - 1
[3,1,1,1]
=> 2 = 3 - 1
[2,2,2]
=> 0 = 1 - 1
[2,2,1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> 0 = 1 - 1
[7]
=> 0 = 1 - 1
[6,1]
=> 5 = 6 - 1
[5,2]
=> 3 = 4 - 1
[5,1,1]
=> 4 = 5 - 1
[4,3]
=> 1 = 2 - 1
[4,2,1]
=> 3 = 4 - 1
[4,1,1,1]
=> 3 = 4 - 1
[3,3,1]
=> 2 = 3 - 1
[3,2,2]
=> 1 = 2 - 1
[3,2,1,1]
=> 2 = 3 - 1
[3,1,1,1,1]
=> 2 = 3 - 1
[2,2,2,1]
=> 1 = 2 - 1
[2,2,1,1,1]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> 0 = 1 - 1
[8]
=> 0 = 1 - 1
[7,1]
=> 6 = 7 - 1
[6,2]
=> 4 = 5 - 1
[6,1,1]
=> 5 = 6 - 1
[5,3]
=> 2 = 3 - 1
[5,2,1]
=> 4 = 5 - 1
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St001804
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001804: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> 1
[2]
=> [[1,2]]
=> 1
[1,1]
=> [[1],[2]]
=> 1
[3]
=> [[1,2,3]]
=> 1
[2,1]
=> [[1,3],[2]]
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> 1
[4]
=> [[1,2,3,4]]
=> 1
[3,1]
=> [[1,3,4],[2]]
=> 2
[2,2]
=> [[1,2],[3,4]]
=> 1
[2,1,1]
=> [[1,4],[2],[3]]
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[5]
=> [[1,2,3,4,5]]
=> 1
[4,1]
=> [[1,3,4,5],[2]]
=> 2
[3,2]
=> [[1,2,5],[3,4]]
=> 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> 2
[4,2]
=> [[1,2,5,6],[3,4]]
=> 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 3
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 3
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 5
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[7]
=> [[1,2,3,4,5,6,7]]
=> 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> 2
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> 2
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 3
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> 2
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 4
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 3
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 4
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 5
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 2
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 4
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 6
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> 2
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> 2
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> 3
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> 2
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> 3
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St000288
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 10 => 1
[2]
=> 100 => 011 => 100 => 1
[1,1]
=> 110 => 001 => 010 => 1
[3]
=> 1000 => 0111 => 1000 => 1
[2,1]
=> 1010 => 0101 => 1010 => 2
[1,1,1]
=> 1110 => 0001 => 0010 => 1
[4]
=> 10000 => 01111 => 10000 => 1
[3,1]
=> 10010 => 01101 => 10110 => 3
[2,2]
=> 1100 => 0011 => 0100 => 1
[2,1,1]
=> 10110 => 01001 => 10010 => 2
[1,1,1,1]
=> 11110 => 00001 => 00010 => 1
[5]
=> 100000 => 011111 => 100000 => 1
[4,1]
=> 100010 => 011101 => 101110 => 4
[3,2]
=> 10100 => 01011 => 10100 => 2
[3,1,1]
=> 100110 => 011001 => 100110 => 3
[2,2,1]
=> 11010 => 00101 => 01010 => 2
[2,1,1,1]
=> 101110 => 010001 => 100010 => 2
[1,1,1,1,1]
=> 111110 => 000001 => 000010 => 1
[6]
=> 1000000 => 0111111 => 1000000 => 1
[5,1]
=> 1000010 => 0111101 => 1011110 => 5
[4,2]
=> 100100 => 011011 => 101100 => 3
[4,1,1]
=> 1000110 => 0111001 => 1001110 => 4
[3,3]
=> 11000 => 00111 => 01000 => 1
[3,2,1]
=> 101010 => 010101 => 101010 => 3
[3,1,1,1]
=> 1001110 => 0110001 => 1000110 => 3
[2,2,2]
=> 11100 => 00011 => 00100 => 1
[2,2,1,1]
=> 110110 => 001001 => 010010 => 2
[2,1,1,1,1]
=> 1011110 => 0100001 => 1000010 => 2
[1,1,1,1,1,1]
=> 1111110 => 0000001 => 0000010 => 1
[7]
=> 10000000 => 01111111 => 10000000 => 1
[6,1]
=> 10000010 => 01111101 => 10111110 => 6
[5,2]
=> 1000100 => 0111011 => 1011100 => 4
[5,1,1]
=> 10000110 => 01111001 => 10011110 => 5
[4,3]
=> 101000 => 010111 => 101000 => 2
[4,2,1]
=> 1001010 => 0110101 => 1011010 => 4
[4,1,1,1]
=> 10001110 => 01110001 => 10001110 => 4
[3,3,1]
=> 110010 => 001101 => 010110 => 3
[3,2,2]
=> 101100 => 010011 => 100100 => 2
[3,2,1,1]
=> 1010110 => 0101001 => 1010010 => 3
[3,1,1,1,1]
=> 10011110 => 01100001 => 10000110 => 3
[2,2,2,1]
=> 111010 => 000101 => 001010 => 2
[2,2,1,1,1]
=> 1101110 => 0010001 => 0100010 => 2
[2,1,1,1,1,1]
=> 10111110 => 01000001 => 10000010 => 2
[1,1,1,1,1,1,1]
=> 11111110 => 00000001 => 00000010 => 1
[8]
=> 100000000 => 011111111 => 100000000 => 1
[7,1]
=> 100000010 => 011111101 => 101111110 => 7
[6,2]
=> 10000100 => 01111011 => 10111100 => 5
[6,1,1]
=> 100000110 => 011111001 => 100111110 => 6
[5,3]
=> 1001000 => 0110111 => 1011000 => 3
[5,2,1]
=> 10001010 => 01110101 => 10111010 => 5
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000010
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> []
=> 0 = 1 - 1
[2]
=> [[2],[]]
=> [[2],[]]
=> []
=> 0 = 1 - 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> []
=> 0 = 1 - 1
[3]
=> [[3],[]]
=> [[3],[]]
=> []
=> 0 = 1 - 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> 1 = 2 - 1
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> 0 = 1 - 1
[4]
=> [[4],[]]
=> [[4],[]]
=> []
=> 0 = 1 - 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> 1 = 2 - 1
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> []
=> 0 = 1 - 1
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[5]
=> [[5],[]]
=> [[5],[]]
=> []
=> 0 = 1 - 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [1]
=> 1 = 2 - 1
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 3 - 1
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [1]
=> 1 = 2 - 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[6]
=> [[6],[]]
=> [[6],[]]
=> []
=> 0 = 1 - 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [4]
=> 1 = 2 - 1
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [2]
=> 1 = 2 - 1
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [3,3]
=> 2 = 3 - 1
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> []
=> 0 = 1 - 1
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 2 = 3 - 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3 = 4 - 1
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> []
=> 0 = 1 - 1
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[7]
=> [[7],[]]
=> [[7],[]]
=> []
=> 0 = 1 - 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [5]
=> 1 = 2 - 1
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [3]
=> 1 = 2 - 1
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [4,4]
=> 2 = 3 - 1
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [1]
=> 1 = 2 - 1
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [3,2]
=> 2 = 3 - 1
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 3 = 4 - 1
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [2]
=> 1 = 2 - 1
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 4 = 5 - 1
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [1]
=> 1 = 2 - 1
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[8]
=> [[8],[]]
=> [[8],[]]
=> []
=> 0 = 1 - 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [6]
=> 1 = 2 - 1
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [4]
=> 1 = 2 - 1
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [5,5]
=> 2 = 3 - 1
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [2]
=> 1 = 2 - 1
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [4,3]
=> 2 = 3 - 1
Description
The length of the partition.
Matching statistic: St000147
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> []
=> 0 = 1 - 1
[2]
=> [[2],[]]
=> [[2],[]]
=> []
=> 0 = 1 - 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> []
=> 0 = 1 - 1
[3]
=> [[3],[]]
=> [[3],[]]
=> []
=> 0 = 1 - 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> 1 = 2 - 1
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> 0 = 1 - 1
[4]
=> [[4],[]]
=> [[4],[]]
=> []
=> 0 = 1 - 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> 2 = 3 - 1
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> []
=> 0 = 1 - 1
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[5]
=> [[5],[]]
=> [[5],[]]
=> []
=> 0 = 1 - 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> 3 = 4 - 1
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [1]
=> 1 = 2 - 1
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 3 - 1
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [1]
=> 1 = 2 - 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[6]
=> [[6],[]]
=> [[6],[]]
=> []
=> 0 = 1 - 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [4]
=> 4 = 5 - 1
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [2]
=> 2 = 3 - 1
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [3,3]
=> 3 = 4 - 1
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> []
=> 0 = 1 - 1
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 2 = 3 - 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2 = 3 - 1
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> []
=> 0 = 1 - 1
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[7]
=> [[7],[]]
=> [[7],[]]
=> []
=> 0 = 1 - 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [5]
=> 5 = 6 - 1
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [3]
=> 3 = 4 - 1
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [4,4]
=> 4 = 5 - 1
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [1]
=> 1 = 2 - 1
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [3,2]
=> 3 = 4 - 1
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 3 = 4 - 1
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [2]
=> 2 = 3 - 1
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 2 = 3 - 1
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [1]
=> 1 = 2 - 1
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
[8]
=> [[8],[]]
=> [[8],[]]
=> []
=> 0 = 1 - 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [6]
=> 6 = 7 - 1
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [4]
=> 4 = 5 - 1
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [5,5]
=> 5 = 6 - 1
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [2]
=> 2 = 3 - 1
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [4,3]
=> 4 = 5 - 1
Description
The largest part of an integer partition.
Matching statistic: St000766
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000766: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000766: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 0 = 1 - 1
[2]
=> [[1,2]]
=> [2] => [2] => 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => [1,1] => 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [3] => [3] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => [2,1] => 1 = 2 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [1,1,1] => 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [4] => [4] => 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => [3,1] => 1 = 2 - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2,2] => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => 2 = 3 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => [5] => 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => [4,1] => 1 = 2 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => [3,2] => 1 = 2 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => [3,1,1] => 2 = 3 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [2,1,2] => 1 = 2 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => [2,1,1,1] => 3 = 4 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [6] => 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => [5,1] => 1 = 2 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => [4,2] => 1 = 2 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => [4,1,1] => 2 = 3 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [3,3] => 0 = 1 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => [3,1,2] => 2 = 3 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => [3,1,1,1] => 3 = 4 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [2,2,2] => 0 = 1 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => [2,1,1,2] => 2 = 3 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => [2,1,1,1,1] => 4 = 5 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => 0 = 1 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [7] => 0 = 1 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => [6,1] => 1 = 2 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => [5,2] => 1 = 2 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => [5,1,1] => 2 = 3 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => [4,3] => 1 = 2 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => [4,1,2] => 2 = 3 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => [4,1,1,1] => 3 = 4 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => [3,1,3] => 1 = 2 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => [3,2,2] => 2 = 3 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => [3,1,1,2] => 3 = 4 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => [3,1,1,1,1] => 4 = 5 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => [2,1,2,2] => 1 = 2 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => [2,1,1,1,2] => 3 = 4 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => [2,1,1,1,1,1] => 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0 = 1 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [8] => 0 = 1 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => [7,1] => 1 = 2 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => [6,2] => 1 = 2 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => [6,1,1] => 2 = 3 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => [5,3] => 1 = 2 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => [5,1,2] => 2 = 3 - 1
Description
The number of inversions of an integer composition.
This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.
Matching statistic: St000052
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 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]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 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]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 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]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 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]
=> [1,1,1,1,0,0,1,0,0,0]
=> 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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 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]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 6 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 4 = 5 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,5} - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 7 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> 5 = 6 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 4 = 5 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {1,5} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000356
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1 = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 1 = 2 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,6,4,5] => 2 = 3 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => 2 = 3 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3 = 4 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => 2 = 3 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,6,2,3,4,5] => 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1 = 2 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [1,2,3,5,6,4] => 1 = 2 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => 2 = 3 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 2 = 3 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => [1,2,4,6,3,5] => 2 = 3 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => 3 = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => 1 = 2 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => [1,3,6,2,4,5] => 3 = 4 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => 4 = 5 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => 1 = 2 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [2,6,1,3,4,5] => 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => 1 = 2 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => 1 = 2 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,6,7] => 2 = 3 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [1,2,5,6,3,4] => 2 = 3 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => 2 = 3 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,8,5,6,7] => ? ∊ {4,6} - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [1,2,8,3,4,5,6,7] => ? ∊ {4,6} - 1
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000996
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 94%●distinct values known / distinct values provided: 86%
Values
[1]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => [1,2,3] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [1,4,2,3] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1 = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,6,4] => 2 = 3 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,5,3] => 2 = 3 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,4,5,6,3] => 3 = 4 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,3,4] => 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [1,4,5,2,3] => 2 = 3 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,3,4,5,6,2] => 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1 = 2 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [1,2,3,6,4,5] => 1 = 2 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => 2 = 3 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 2 = 3 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => [1,2,5,3,6,4] => 2 = 3 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => 3 = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [1,5,2,3,4] => 1 = 2 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => [1,4,2,5,6,3] => 3 = 4 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => 4 = 5 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [1,2,5,3,4] => 1 = 2 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [1,4,5,6,2,3] => 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,3,4,5,6,7,2] => 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => 1 = 2 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => 1 = 2 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [1,2,3,4,5,7,8,6] => ? ∊ {3,4,6,7} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [1,2,5,6,3,4] => 2 = 3 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => 2 = 3 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,8,5] => ? ∊ {3,4,6,7} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [1,2,4,5,6,7,8,3] => ? ∊ {3,4,6,7} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,8,2] => ? ∊ {3,4,6,7} - 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000204
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 86% ●values known / values provided: 88%●distinct values known / distinct values provided: 86%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 86% ●values known / values provided: 88%●distinct values known / distinct values provided: 86%
Values
[1]
=> [1,0]
=> [1,0]
=> [.,.]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 3 = 4 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 4 = 5 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 2 = 3 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> 5 = 6 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 3 = 4 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> 4 = 5 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 1 = 2 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 1 = 2 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[[.,[.,.]],.],[[.,.],.]]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2 = 3 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? ∊ {1,1,2,3,3,5,6,7} - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? ∊ {1,1,2,3,3,5,6,7} - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> 4 = 5 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? ∊ {1,1,2,3,3,5,6,7} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,.],[.,[[[.,.],.],.]]]
=> 2 = 3 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[[[[.,.],.],.],.]]
=> 4 = 5 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? ∊ {1,1,2,3,3,5,6,7} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 0 = 1 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> 2 = 3 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> 2 = 3 - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? ∊ {1,1,2,3,3,5,6,7} - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? ∊ {1,1,2,3,3,5,6,7} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? ∊ {1,1,2,3,3,5,6,7} - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? ∊ {1,1,2,3,3,5,6,7} - 1
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
The following 181 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000446The disorder of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000496The rcs statistic of a set partition. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000358The number of occurrences of the pattern 31-2. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000317The cycle descent number of a permutation. St000836The number of descents of distance 2 of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000710The number of big deficiencies of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St001083The number of boxed occurrences of 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000451The length of the longest pattern of the form k 1 2. St001323The independence gap of a graph. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000993The multiplicity of the largest part of an integer partition. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000886The number of permutations with the same antidiagonal sums. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St000355The number of occurrences of the pattern 21-3. St000365The number of double ascents of a permutation. St000462The major index minus the number of excedences of a permutation. St001513The number of nested exceedences of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001689The number of celebrities in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001435The number of missing boxes in the first row. St001964The interval resolution global dimension of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000939The number of characters of the symmetric group whose value on the partition is positive. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001867The number of alignments of type EN of a signed permutation. St000422The energy of a graph, if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001487The number of inner corners of a skew partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001438The number of missing boxes of a skew partition. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St000177The number of free tiles in the pattern. St000178Number of free entries. St000454The largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice. St000456The monochromatic index of a connected graph. St000769The major index of a composition regarded as a word. St001820The size of the image of the pop stack sorting operator. St001556The number of inversions of the third entry of a permutation. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St000299The number of nonisomorphic vertex-induced subtrees. St000516The number of stretching pairs of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001520The number of strict 3-descents. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001822The number of alignments of a signed permutation. St000527The width of the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000455The second largest eigenvalue of a graph if it is integral. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St000902 The minimal number of repetitions of an integer composition. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000488The number of cycles of a permutation of length at most 2. St000630The length of the shortest palindromic decomposition of a binary word. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000765The number of weak records in an integer composition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001530The depth of a Dyck path. St001856The number of edges in the reduced word graph of a permutation. St001868The number of alignments of type NE of a signed permutation. St001948The number of augmented double ascents of a permutation. St001866The nesting alignments of a signed permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. St000068The number of minimal elements in a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000679The pruning number of an ordered tree. St001084The number of occurrences of the vincular pattern |1-23 in 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 $eA$. St001199The dominant dimension of $eAe$ for 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 $eA$. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001896The number of right descents of a signed permutations. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000338The number of pixed points of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000779The tier of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000871The number of very big ascents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001490The number of connected components of a skew 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. St000022The number of fixed points of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000862The number of parts of the shifted shape of a permutation. St001488The number of corners of a skew partition.
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