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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 1
[2]
=> []
=> 1
[1,1]
=> [1]
=> 2
[3]
=> []
=> 1
[2,1]
=> [1]
=> 2
[1,1,1]
=> [1,1]
=> 3
[4]
=> []
=> 1
[3,1]
=> [1]
=> 2
[2,2]
=> [2]
=> 3
[2,1,1]
=> [1,1]
=> 3
[1,1,1,1]
=> [1,1,1]
=> 4
[5]
=> []
=> 1
[4,1]
=> [1]
=> 2
[3,2]
=> [2]
=> 3
[3,1,1]
=> [1,1]
=> 3
[2,2,1]
=> [2,1]
=> 5
[2,1,1,1]
=> [1,1,1]
=> 4
[1,1,1,1,1]
=> [1,1,1,1]
=> 5
[6]
=> []
=> 1
[5,1]
=> [1]
=> 2
[4,2]
=> [2]
=> 3
[4,1,1]
=> [1,1]
=> 3
[3,3]
=> [3]
=> 4
[3,2,1]
=> [2,1]
=> 5
[3,1,1,1]
=> [1,1,1]
=> 4
[2,2,2]
=> [2,2]
=> 6
[2,2,1,1]
=> [2,1,1]
=> 7
[2,1,1,1,1]
=> [1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[7]
=> []
=> 1
[6,1]
=> [1]
=> 2
[5,2]
=> [2]
=> 3
[5,1,1]
=> [1,1]
=> 3
[4,3]
=> [3]
=> 4
[4,2,1]
=> [2,1]
=> 5
[4,1,1,1]
=> [1,1,1]
=> 4
[3,3,1]
=> [3,1]
=> 7
[3,2,2]
=> [2,2]
=> 6
[3,2,1,1]
=> [2,1,1]
=> 7
[3,1,1,1,1]
=> [1,1,1,1]
=> 5
[2,2,2,1]
=> [2,2,1]
=> 9
[2,2,1,1,1]
=> [2,1,1,1]
=> 9
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[6,2]
=> [2]
=> 3
[5,3]
=> [3]
=> 4
[5,2,1]
=> [2,1]
=> 5
[4,4]
=> [4]
=> 5
[4,3,1]
=> [3,1]
=> 7
[4,2,2]
=> [2,2]
=> 6
[4,2,1,1]
=> [2,1,1]
=> 7
Description
The number of partitions contained in the given partition.
Matching statistic: St001389
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 2
[3]
=> [1,1,1]
=> 1
[2,1]
=> [2,1]
=> 2
[1,1,1]
=> [3]
=> 3
[4]
=> [1,1,1,1]
=> 1
[3,1]
=> [2,1,1]
=> 2
[2,2]
=> [2,2]
=> 3
[2,1,1]
=> [3,1]
=> 3
[1,1,1,1]
=> [4]
=> 4
[5]
=> [1,1,1,1,1]
=> 1
[4,1]
=> [2,1,1,1]
=> 2
[3,2]
=> [2,2,1]
=> 3
[3,1,1]
=> [3,1,1]
=> 3
[2,2,1]
=> [3,2]
=> 5
[2,1,1,1]
=> [4,1]
=> 4
[1,1,1,1,1]
=> [5]
=> 5
[6]
=> [1,1,1,1,1,1]
=> 1
[5,1]
=> [2,1,1,1,1]
=> 2
[4,2]
=> [2,2,1,1]
=> 3
[4,1,1]
=> [3,1,1,1]
=> 3
[3,3]
=> [2,2,2]
=> 4
[3,2,1]
=> [3,2,1]
=> 5
[3,1,1,1]
=> [4,1,1]
=> 4
[2,2,2]
=> [3,3]
=> 6
[2,2,1,1]
=> [4,2]
=> 7
[2,1,1,1,1]
=> [5,1]
=> 5
[1,1,1,1,1,1]
=> [6]
=> 6
[7]
=> [1,1,1,1,1,1,1]
=> 1
[6,1]
=> [2,1,1,1,1,1]
=> 2
[5,2]
=> [2,2,1,1,1]
=> 3
[5,1,1]
=> [3,1,1,1,1]
=> 3
[4,3]
=> [2,2,2,1]
=> 4
[4,2,1]
=> [3,2,1,1]
=> 5
[4,1,1,1]
=> [4,1,1,1]
=> 4
[3,3,1]
=> [3,2,2]
=> 7
[3,2,2]
=> [3,3,1]
=> 6
[3,2,1,1]
=> [4,2,1]
=> 7
[3,1,1,1,1]
=> [5,1,1]
=> 5
[2,2,2,1]
=> [4,3]
=> 9
[2,2,1,1,1]
=> [5,2]
=> 9
[2,1,1,1,1,1]
=> [6,1]
=> 6
[6,2]
=> [2,2,1,1,1,1]
=> 3
[5,3]
=> [2,2,2,1,1]
=> 4
[5,2,1]
=> [3,2,1,1,1]
=> 5
[4,4]
=> [2,2,2,2]
=> 5
[4,3,1]
=> [3,2,2,1]
=> 7
[4,2,2]
=> [3,3,1,1]
=> 6
[4,2,1,1]
=> [4,2,1,1]
=> 7
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001313
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00104: Binary words —reverse⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001313: 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 => 001 => 100 => 1
[1,1]
=> 110 => 011 => 101 => 2
[3]
=> 1000 => 0001 => 1000 => 1
[2,1]
=> 1010 => 0101 => 1010 => 2
[1,1,1]
=> 1110 => 0111 => 1011 => 3
[4]
=> 10000 => 00001 => 10000 => 1
[3,1]
=> 10010 => 01001 => 10100 => 2
[2,2]
=> 1100 => 0011 => 1001 => 3
[2,1,1]
=> 10110 => 01101 => 10110 => 3
[1,1,1,1]
=> 11110 => 01111 => 10111 => 4
[5]
=> 100000 => 000001 => 100000 => 1
[4,1]
=> 100010 => 010001 => 101000 => 2
[3,2]
=> 10100 => 00101 => 10010 => 3
[3,1,1]
=> 100110 => 011001 => 101100 => 3
[2,2,1]
=> 11010 => 01011 => 10101 => 5
[2,1,1,1]
=> 101110 => 011101 => 101110 => 4
[1,1,1,1,1]
=> 111110 => 011111 => 101111 => 5
[6]
=> 1000000 => 0000001 => 1000000 => 1
[5,1]
=> 1000010 => 0100001 => 1010000 => 2
[4,2]
=> 100100 => 001001 => 100100 => 3
[4,1,1]
=> 1000110 => 0110001 => 1011000 => 3
[3,3]
=> 11000 => 00011 => 10001 => 4
[3,2,1]
=> 101010 => 010101 => 101010 => 5
[3,1,1,1]
=> 1001110 => 0111001 => 1011100 => 4
[2,2,2]
=> 11100 => 00111 => 10011 => 6
[2,2,1,1]
=> 110110 => 011011 => 101101 => 7
[2,1,1,1,1]
=> 1011110 => 0111101 => 1011110 => 5
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 1011111 => 6
[7]
=> 10000000 => 00000001 => 10000000 => 1
[6,1]
=> 10000010 => 01000001 => 10100000 => 2
[5,2]
=> 1000100 => 0010001 => 1001000 => 3
[5,1,1]
=> 10000110 => 01100001 => 10110000 => 3
[4,3]
=> 101000 => 000101 => 100010 => 4
[4,2,1]
=> 1001010 => 0101001 => 1010100 => 5
[4,1,1,1]
=> 10001110 => 01110001 => 10111000 => 4
[3,3,1]
=> 110010 => 010011 => 101001 => 7
[3,2,2]
=> 101100 => 001101 => 100110 => 6
[3,2,1,1]
=> 1010110 => 0110101 => 1011010 => 7
[3,1,1,1,1]
=> 10011110 => 01111001 => 10111100 => 5
[2,2,2,1]
=> 111010 => 010111 => 101011 => 9
[2,2,1,1,1]
=> 1101110 => 0111011 => 1011101 => 9
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 10111110 => 6
[6,2]
=> 10000100 => 00100001 => 10010000 => 3
[5,3]
=> 1001000 => 0001001 => 1000100 => 4
[5,2,1]
=> 10001010 => 01010001 => 10101000 => 5
[4,4]
=> 110000 => 000011 => 100001 => 5
[4,3,1]
=> 1010010 => 0100101 => 1010010 => 7
[4,2,2]
=> 1001100 => 0011001 => 1001100 => 6
[4,2,1,1]
=> 10010110 => 01101001 => 10110100 => 7
Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St001464
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 5
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 4
[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]
=> [2,3,4,5,1] => 5
[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]
=> [1,2,3,4,5,6] => 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]
=> [1,2,3,4,6,5] => 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 3
[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]
=> [1,2,3,5,6,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 5
[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]
=> [1,2,4,5,6,3] => 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 6
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 7
[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]
=> [1,3,4,5,6,2] => 5
[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]
=> [2,3,4,5,6,1] => 6
[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]
=> [1,2,3,4,5,6,7] => 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]
=> [1,2,3,4,5,7,6] => 2
[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]
=> [1,2,3,6,4,5] => 3
[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]
=> [1,2,3,4,6,7,5] => 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 4
[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]
=> [1,2,5,3,6,4] => 5
[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]
=> [1,2,3,5,6,7,4] => 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 7
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 6
[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]
=> [1,4,2,5,6,3] => 7
[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]
=> [1,2,4,5,6,7,3] => 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 9
[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]
=> [3,1,4,5,6,2] => 9
[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]
=> [1,3,4,5,6,7,2] => 6
[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]
=> [1,2,3,4,7,5,6] => 3
[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]
=> [1,2,6,3,4,5] => 4
[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]
=> [1,2,3,6,4,7,5] => 5
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 5
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => 7
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,2,5,3,6,7,4] => 7
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St000420
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 93%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 93%
Values
[1]
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> 2
[3]
=> []
=> []
=> ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4]
=> []
=> []
=> ? = 1
[3,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[5]
=> []
=> []
=> ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> 2
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[6]
=> []
=> []
=> ? = 1
[5,1]
=> [1]
=> [1,0,1,0]
=> 2
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[7]
=> []
=> []
=> ? = 1
[6,1]
=> [1]
=> [1,0,1,0]
=> 2
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 7
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 9
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 9
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 7
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 9
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 9
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 9
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 10
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 12
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 93%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 93%
Values
[1]
=> []
=> []
=> ? = 1 - 1
[2]
=> []
=> []
=> ? = 1 - 1
[1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[3]
=> []
=> []
=> ? = 1 - 1
[2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[4]
=> []
=> []
=> ? = 1 - 1
[3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[5]
=> []
=> []
=> ? = 1 - 1
[4,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[6]
=> []
=> []
=> ? = 1 - 1
[5,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[7]
=> []
=> []
=> ? = 1 - 1
[6,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 8 = 9 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 8 = 9 - 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 9 = 10 - 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 11 = 12 - 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St000070
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 93%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 93%
Values
[1]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[2]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[3]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[2,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[1,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[4]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[3,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[2,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[2,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[5]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[4,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[3,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[3,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[2,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[6]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[5,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[4,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[4,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[3,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
[3,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[2,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6
[7]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[6,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[5,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[5,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[4,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
[4,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[4,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[3,3,1]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[3,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 9
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6
[6,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[5,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
[5,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[4,4]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[4,3,1]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[4,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[4,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[3,3,2]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[3,3,1,1]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 10
[3,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[3,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 9
[2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 10
[2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 12
[6,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
Description
The number of antichains in a poset.
An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable.
An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St000110
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 5
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 4
[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]
=> [2,3,4,5,1] => 5
[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]
=> [1,2,3,4,5,6] => 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]
=> [1,2,3,4,6,5] => 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 3
[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]
=> [1,2,3,5,6,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 5
[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]
=> [1,2,4,5,6,3] => 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 6
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 7
[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]
=> [1,3,4,5,6,2] => 5
[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]
=> [2,3,4,5,6,1] => 6
[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]
=> [1,2,3,4,5,6,7] => 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]
=> [1,2,3,4,5,7,6] => 2
[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]
=> [1,2,3,6,4,5] => 3
[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]
=> [1,2,3,4,6,7,5] => ? = 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 4
[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]
=> [1,2,5,3,6,4] => 5
[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]
=> [1,2,3,5,6,7,4] => ? = 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 7
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 6
[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]
=> [1,4,2,5,6,3] => 7
[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]
=> [1,2,4,5,6,7,3] => ? = 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 9
[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]
=> [3,1,4,5,6,2] => 9
[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]
=> [1,3,4,5,6,7,2] => ? = 6
[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]
=> [1,2,3,4,7,5,6] => ? = 3
[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]
=> [1,2,6,3,4,5] => 4
[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]
=> [1,2,3,6,4,7,5] => 5
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 5
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => 7
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,2,5,3,6,7,4] => ? = 7
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 9
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,2,5,6,3] => 10
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,2,6,3] => 9
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,2,5,6,7,3] => ? = 9
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 10
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => 12
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 4
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => 5
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,2,3,6,7,4,5] => ? = 6
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,2,5,6,3,7,4] => ? = 9
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,5,2,6,7,3] => ? = 12
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 5
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,2,6,3,7,4,5] => ? = 9
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,3,4] => ? = 10
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,4,5,6,2,7,3] => ? = 14
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,2,6,7,3,4,5] => ? = 10
[3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,2,3] => ? = 15
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000883
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 53%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 53%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => 3
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => 3
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 3
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => 5
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 4
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => 3
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 4
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,4,2,6,5,3] => 5
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => 6
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => 7
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 5
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 2
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,2,3,6,4,7,5] => 3
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 3
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,5,2,6,3,7,4] => ? = 4
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,5,3,7,6,4] => 5
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 4
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [4,1,5,2,7,6,3] => ? = 7
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,4,2,7,5,3] => ? = 6
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,4,2,7,6,5,3] => 7
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 5
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [5,3,1,7,6,4,2] => ? = 9
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [3,1,7,6,5,4,2] => ? = 9
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 6
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [1,2,3,4,7,5,8,6] => ? = 3
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,8,5] => ? = 4
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,6,4,8,7,5] => ? = 5
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 5
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,5,2,6,3,8,7,4] => ? = 7
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,2,7,5,3,8,6,4] => ? = 6
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,5,3,8,7,6,4] => ? = 7
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [4,1,7,5,2,8,6,3] => ? = 9
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [4,1,5,2,8,7,6,3] => ? = 10
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,6,4,2,8,7,5,3] => ? = 9
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,4,2,8,7,6,5,3] => ? = 9
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 10
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [5,3,1,8,7,6,4,2] => ? = 12
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [1,2,3,7,4,8,5,9,6] => ? = 4
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [1,6,2,7,3,8,4,9,5] => ? = 5
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,2,6,3,7,4,9,8,5] => ? = 7
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [1,2,3,8,6,4,9,7,5] => ? = 6
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,9,8,4] => ? = 9
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [1,5,2,8,6,3,9,7,4] => ? = 9
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [1,2,7,5,3,9,8,6,4] => ? = 9
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [7,4,1,8,5,2,9,6,3] => ? = 10
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [4,1,7,5,2,9,8,6,3] => ? = 14
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [1,8,6,4,2,9,7,5,3] => ? = 10
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [1,6,4,2,9,8,7,5,3] => ? = 12
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [7,5,3,1,9,8,6,4,2] => ? = 14
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => [1,2,7,3,8,4,9,5,10,6] => ? = 5
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [6,1,7,2,8,3,9,4,10,5] => ? = 6
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [9,10,6,7,8,1,2,3,4,5] => [1,2,6,3,9,7,4,10,8,5] => ? = 9
[4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [9,10,5,6,7,8,1,2,3,4] => [5,1,6,2,9,7,3,10,8,4] => ? = 12
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [8,9,10,5,6,7,1,2,3,4] => [1,8,5,2,9,6,3,10,7,4] => ? = 10
[4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,1,2,3,4] => [1,2,9,7,5,3,10,8,6,4] => ? = 10
[3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [10,7,8,9,4,5,6,1,2,3] => [7,4,1,8,5,2,10,9,6,3] => ? = 16
[3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [9,10,7,8,4,5,6,1,2,3] => [4,1,9,7,5,2,10,8,6,3] => ? = 16
[3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,4,5,1,2,3] => [1,8,6,4,2,10,9,7,5,3] => ? = 14
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => [9,7,5,3,1,10,8,6,4,2] => ? = 15
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [9,10,11,6,7,8,1,2,3,4,5] => [1,2,9,6,3,10,7,4,11,8,5] => ? = 10
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => [5,1,9,6,2,10,7,3,11,8,4] => ? = 14
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => [7,4,1,10,8,5,2,11,9,6,3] => ? = 19
[3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? => ? = 15
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? => ? = 15
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? => ? = 20
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001684
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 60%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 60%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 3 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3 = 4 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 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]
=> [1,2,3,5,4] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2 = 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]
=> [1,2,4,5,3] => 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 4 = 5 - 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]
=> [1,3,4,5,2] => 3 = 4 - 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]
=> [2,3,4,5,1] => 4 = 5 - 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]
=> [1,2,3,4,5,6] => ? = 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]
=> [1,2,3,4,6,5] => ? = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 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]
=> [1,2,3,5,6,4] => ? = 3 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3 = 4 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 4 = 5 - 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]
=> [1,2,4,5,6,3] => ? = 4 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 5 = 6 - 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]
=> [3,1,4,5,2] => 6 = 7 - 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]
=> [1,3,4,5,6,2] => ? = 5 - 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]
=> [2,3,4,5,6,1] => ? = 6 - 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]
=> [1,2,3,4,5,6,7] => ? = 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]
=> [1,2,3,4,5,7,6] => ? = 2 - 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]
=> [1,2,3,6,4,5] => ? = 3 - 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]
=> [1,2,3,4,6,7,5] => ? = 3 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 3 = 4 - 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]
=> [1,2,5,3,6,4] => ? = 5 - 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]
=> [1,2,3,5,6,7,4] => ? = 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]
=> [4,1,2,5,3] => 6 = 7 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 5 = 6 - 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]
=> [1,4,2,5,6,3] => ? = 7 - 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]
=> [1,2,4,5,6,7,3] => ? = 5 - 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]
=> [3,4,1,5,2] => 8 = 9 - 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]
=> [3,1,4,5,6,2] => ? = 9 - 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]
=> [1,3,4,5,6,7,2] => ? = 6 - 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]
=> [1,2,3,4,7,5,6] => ? = 3 - 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]
=> [1,2,6,3,4,5] => ? = 4 - 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]
=> [1,2,3,6,4,7,5] => ? = 5 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 4 = 5 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => ? = 7 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => ? = 6 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,2,5,3,6,7,4] => ? = 7 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 8 = 9 - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,2,5,6,3] => ? = 10 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,5,2,6,3] => ? = 9 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,2,5,6,7,3] => ? = 9 - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 9 = 10 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => ? = 12 - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 4 - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => ? = 5 - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,2,6,3,4,7,5] => ? = 7 - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,2,3,6,7,4,5] => ? = 6 - 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => ? = 9 - 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => ? = 9 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,2,5,6,3,7,4] => ? = 9 - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 9 = 10 - 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3] => ? = 14 - 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => ? = 10 - 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,5,2,6,7,3] => ? = 12 - 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ? = 14 - 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 5 - 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => ? = 6 - 1
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,2,6,3,7,4,5] => ? = 9 - 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => ? = 12 - 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => ? = 10 - 1
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,3,4] => ? = 10 - 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => ? = 16 - 1
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => ? = 16 - 1
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,4,5,6,2,7,3] => ? = 14 - 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => ? = 15 - 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,2,6,7,3,4,5] => ? = 10 - 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => ? = 14 - 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => ? = 19 - 1
Description
The reduced word complexity of a permutation.
For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$.
For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$.
This statistic appears in [1, Question 6.1].
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001726The number of visible inversions of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001596The number of two-by-two squares inside a skew partition. St000141The maximum drop size of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000662The staircase size of the code of a permutation.
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