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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000110
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: 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]
=> [2,1] => 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[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]
=> [1,2,3,4,5] => 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]
=> [6,1,2,3,4,5] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 7
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 6
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 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]
=> [3,1,2,4,5,6] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
[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]
=> [1,2,3,4,5,6] => 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]
=> [7,1,2,3,4,5,6] => 7
[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]
=> [6,1,2,3,4,5,7] => 6
[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]
=> [5,1,2,3,6,4] => 9
[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]
=> [5,1,2,3,4,6,7] => 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 9
[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]
=> [4,1,2,5,3,6] => 7
[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]
=> [4,1,2,3,5,6,7] => 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 6
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 7
[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]
=> [3,1,4,2,5,6] => 5
[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]
=> [3,1,2,4,5,6,7] => 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[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]
=> [2,3,1,4,5,6] => 3
[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]
=> [2,1,3,4,5,6,7] => 2
[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]
=> [1,2,3,4,5,6,7] => 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]
=> [8,1,2,3,4,5,6,7] => 8
[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]
=> [7,1,2,3,4,5,6,8] => 7
[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]
=> [6,1,2,3,4,7,5] => 11
[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]
=> [6,1,2,3,4,5,7,8] => 6
[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]
=> [5,1,2,6,3,4] => 12
[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]
=> [5,1,2,3,6,4,7] => 9
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: St001313
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ 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
Mp00135: Binary words —rotate front-to-back⟶ 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 => 010 => 2
[1,1]
=> 110 => 011 => 110 => 1
[3]
=> 1000 => 0001 => 0010 => 3
[2,1]
=> 1010 => 0101 => 1010 => 2
[1,1,1]
=> 1110 => 0111 => 1110 => 1
[4]
=> 10000 => 00001 => 00010 => 4
[3,1]
=> 10010 => 01001 => 10010 => 3
[2,2]
=> 1100 => 0011 => 0110 => 3
[2,1,1]
=> 10110 => 01101 => 11010 => 2
[1,1,1,1]
=> 11110 => 01111 => 11110 => 1
[5]
=> 100000 => 000001 => 000010 => 5
[4,1]
=> 100010 => 010001 => 100010 => 4
[3,2]
=> 10100 => 00101 => 01010 => 5
[3,1,1]
=> 100110 => 011001 => 110010 => 3
[2,2,1]
=> 11010 => 01011 => 10110 => 3
[2,1,1,1]
=> 101110 => 011101 => 111010 => 2
[1,1,1,1,1]
=> 111110 => 011111 => 111110 => 1
[6]
=> 1000000 => 0000001 => 0000010 => 6
[5,1]
=> 1000010 => 0100001 => 1000010 => 5
[4,2]
=> 100100 => 001001 => 010010 => 7
[4,1,1]
=> 1000110 => 0110001 => 1100010 => 4
[3,3]
=> 11000 => 00011 => 00110 => 6
[3,2,1]
=> 101010 => 010101 => 101010 => 5
[3,1,1,1]
=> 1001110 => 0111001 => 1110010 => 3
[2,2,2]
=> 11100 => 00111 => 01110 => 4
[2,2,1,1]
=> 110110 => 011011 => 110110 => 3
[2,1,1,1,1]
=> 1011110 => 0111101 => 1111010 => 2
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 1111110 => 1
[7]
=> 10000000 => 00000001 => 00000010 => 7
[6,1]
=> 10000010 => 01000001 => 10000010 => 6
[5,2]
=> 1000100 => 0010001 => 0100010 => 9
[5,1,1]
=> 10000110 => 01100001 => 11000010 => 5
[4,3]
=> 101000 => 000101 => 001010 => 9
[4,2,1]
=> 1001010 => 0101001 => 1010010 => 7
[4,1,1,1]
=> 10001110 => 01110001 => 11100010 => 4
[3,3,1]
=> 110010 => 010011 => 100110 => 6
[3,2,2]
=> 101100 => 001101 => 011010 => 7
[3,2,1,1]
=> 1010110 => 0110101 => 1101010 => 5
[3,1,1,1,1]
=> 10011110 => 01111001 => 11110010 => 3
[2,2,2,1]
=> 111010 => 010111 => 101110 => 4
[2,2,1,1,1]
=> 1101110 => 0111011 => 1110110 => 3
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 11111010 => 2
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 11111110 => 1
[8]
=> 100000000 => 000000001 => 000000010 => 8
[7,1]
=> 100000010 => 010000001 => 100000010 => 7
[6,2]
=> 10000100 => 00100001 => 01000010 => 11
[6,1,1]
=> 100000110 => 011000001 => 110000010 => 6
[5,3]
=> 1001000 => 0001001 => 0010010 => 12
[5,2,1]
=> 10001010 => 01010001 => 10100010 => 9
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: St001389
St001389: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 97%●distinct values known / distinct values provided: 92%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 3
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 5
[3,1,1]
=> 3
[2,2,1]
=> 3
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 7
[4,1,1]
=> 4
[3,3]
=> 6
[3,2,1]
=> 5
[3,1,1,1]
=> 3
[2,2,2]
=> 4
[2,2,1,1]
=> 3
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 7
[6,1]
=> 6
[5,2]
=> 9
[5,1,1]
=> 5
[4,3]
=> 9
[4,2,1]
=> 7
[4,1,1,1]
=> 4
[3,3,1]
=> 6
[3,2,2]
=> 7
[3,2,1,1]
=> 5
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 4
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 7
[6,2]
=> 11
[6,1,1]
=> 6
[5,3]
=> 12
[5,2,1]
=> 9
[3,3,3,3,3,3]
=> ? = 28
[5,5,5,5]
=> ? = 70
[6,6,6]
=> ? = 56
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: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 94%●distinct values known / distinct values provided: 85%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 94%●distinct values known / distinct values provided: 85%
Values
[1]
=> [1]
=> []
=> 1
[2]
=> [1,1]
=> [1]
=> 2
[1,1]
=> [2]
=> []
=> 1
[3]
=> [1,1,1]
=> [1,1]
=> 3
[2,1]
=> [2,1]
=> [1]
=> 2
[1,1,1]
=> [3]
=> []
=> 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 4
[3,1]
=> [2,1,1]
=> [1,1]
=> 3
[2,2]
=> [2,2]
=> [2]
=> 3
[2,1,1]
=> [3,1]
=> [1]
=> 2
[1,1,1,1]
=> [4]
=> []
=> 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 4
[3,2]
=> [2,2,1]
=> [2,1]
=> 5
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 3
[2,2,1]
=> [3,2]
=> [2]
=> 3
[2,1,1,1]
=> [4,1]
=> [1]
=> 2
[1,1,1,1,1]
=> [5]
=> []
=> 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 5
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 7
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 4
[3,3]
=> [2,2,2]
=> [2,2]
=> 6
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 5
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 3
[2,2,2]
=> [3,3]
=> [3]
=> 4
[2,2,1,1]
=> [4,2]
=> [2]
=> 3
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 2
[1,1,1,1,1,1]
=> [6]
=> []
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 7
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 9
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 5
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 9
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 7
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 4
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 6
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 7
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 5
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 3
[2,2,2,1]
=> [4,3]
=> [3]
=> 4
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 3
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> []
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 8
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 7
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 11
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 12
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 9
[7,7]
=> [2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> ? = 28
[5,5,5]
=> [3,3,3,3,3]
=> [3,3,3,3]
=> ? = 35
[5,5,5,1]
=> [4,3,3,3,3]
=> [3,3,3,3]
=> ? = 35
[3,3,3,3,3,3]
=> [6,6,6]
=> [6,6]
=> ? = 28
[5,5,5,5]
=> [4,4,4,4,4]
=> [4,4,4,4]
=> ? = 70
[6,6,6]
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> ? = 56
Description
The number of partitions contained in the given partition.
Matching statistic: St000420
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 89%●distinct values known / distinct values provided: 88%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 89%●distinct values known / distinct values provided: 88%
Values
[1]
=> [1]
=> []
=> []
=> ? = 1
[2]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1]
=> [2]
=> []
=> []
=> ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,1]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 9
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 9
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 7
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 8
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 11
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 12
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 9
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 10
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 9
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 9
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 7
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1
[2,2,2,2,2,2,2]
=> [7,7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[3,3,3,3,3,3]
=> [6,6,6]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 28
[5,5,5,5]
=> [4,4,4,4,4]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 70
[6,6,6]
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 56
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 89%●distinct values known / distinct values provided: 88%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 89%●distinct values known / distinct values provided: 88%
Values
[1]
=> [1]
=> []
=> []
=> ? = 1 - 1
[2]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> []
=> ? = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1 - 1
[6]
=> [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
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1 - 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[6,1]
=> [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
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 8 = 9 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7 = 8 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 10 = 11 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 11 = 12 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 8 = 9 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 9 = 10 - 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1 - 1
[2,2,2,2,2,2,2]
=> [7,7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[3,3,3,3,3,3]
=> [6,6,6]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 28 - 1
[5,5,5,5]
=> [4,4,4,4,4]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 70 - 1
[6,6,6]
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 56 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St001464
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 59%●distinct values known / distinct values provided: 58%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 59%●distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[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]
=> [1,2,3,4,5] => 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]
=> [6,1,2,3,4,5] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 7
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 6
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 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]
=> [3,1,2,4,5,6] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
[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]
=> [1,2,3,4,5,6] => 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]
=> [7,1,2,3,4,5,6] => ? = 7
[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]
=> [6,1,2,3,4,5,7] => ? = 6
[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]
=> [5,1,2,3,6,4] => 9
[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]
=> [5,1,2,3,4,6,7] => ? = 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 9
[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]
=> [4,1,2,5,3,6] => 7
[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]
=> [4,1,2,3,5,6,7] => ? = 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 6
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 7
[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]
=> [3,1,4,2,5,6] => 5
[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]
=> [3,1,2,4,5,6,7] => ? = 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[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]
=> [2,3,1,4,5,6] => 3
[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]
=> [2,1,3,4,5,6,7] => ? = 2
[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]
=> [1,2,3,4,5,6,7] => 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]
=> [8,1,2,3,4,5,6,7] => ? = 8
[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]
=> [7,1,2,3,4,5,6,8] => ? = 7
[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]
=> [6,1,2,3,4,7,5] => ? = 11
[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]
=> [6,1,2,3,4,5,7,8] => ? = 6
[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]
=> [5,1,2,6,3,4] => 12
[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]
=> [5,1,2,3,6,4,7] => ? = 9
[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]
=> [5,1,2,3,4,6,7,8] => ? = 5
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 10
[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]
=> [4,1,5,2,3,6] => 9
[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]
=> [4,1,2,5,6,3] => 10
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 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]
=> [3,4,1,2,5,6] => 6
[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]
=> [3,1,4,5,2,6] => 7
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[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]
=> [2,3,4,1,5,6] => 4
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => ? = 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => ? = 2
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 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]
=> [6,1,2,3,7,4,5] => ? = 15
[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]
=> [5,1,6,2,3,4] => 14
[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]
=> [5,1,2,6,3,4,7] => ? = 12
[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]
=> [5,1,2,3,6,7,4] => ? = 13
[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]
=> [4,5,1,2,3,6] => 10
[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]
=> [4,1,5,2,6,3] => 14
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,1,2,5,6,7] => ? = 6
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => ? = 4
[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]
=> [6,1,2,7,3,4,5] => ? = 18
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => ? = 14
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,2,5,6,7,8] => ? = 6
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,1,6,7] => ? = 5
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => ? = 15
[5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,1,6,2,3,7,4] => ? = 23
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,1,7] => ? = 6
[2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,5,1,6,7,8] => ? = 5
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? = 21
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,1,2,3,7,4] => ? = 25
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,1,6,2,3,7] => ? = 19
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 7
[2,2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,1,7,8] => ? = 6
[6,6,1]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => ? = 21
[4,4,4,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,1,2,3,7] => ? = 20
[2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,1,8] => ? = 7
[7,7]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => ? = 28
[4,4,4,1,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,6,1,2,3,7,8] => ? = 20
[2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 8
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 35
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,1,2] => ? = 21
[5,5,5,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,1,2,3,4,8] => ? = 35
[3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,8,1,2] => ? = 28
[5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => ? = 70
[6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,1,2,3,4,5] => ? = 56
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St000883
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 42%●distinct values known / distinct values provided: 31%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 42%●distinct values known / distinct values provided: 31%
Values
[1]
=> [[1]]
=> [[1]]
=> [1] => 1
[2]
=> [[1,2]]
=> [[1],[2]]
=> [2,1] => 2
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 1
[3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 3
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => 7
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => 6
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => 5
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 4
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 3
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => 6
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[1,6],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6] => ? = 9
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 5
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => ? = 9
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ? = 7
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 4
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ? = 6
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? = 7
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => ? = 5
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => 3
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => ? = 4
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => 7
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => ? = 11
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => 6
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[1,6],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6] => ? = 12
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,6,8],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8] => ? = 9
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => 5
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5] => ? = 10
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,5,8],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8] => ? = 9
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,5,7],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7] => ? = 10
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,4,7],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7] => ? = 9
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => ? = 6
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,4,6,8],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8] => ? = 7
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => 5
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => ? = 4
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => ? = 3
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => 2
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [[1,7],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7] => ? = 15
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [[1,6],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6] => ? = 14
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,6,9],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6,9] => ? = 12
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,6,8],[2,7,9],[3],[4],[5]]
=> [5,4,3,2,7,9,1,6,8] => ? = 13
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5,9] => ? = 10
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> [4,3,7,2,6,9,1,5,8] => ? = 14
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> [3,6,9,2,5,8,1,4,7] => ? = 10
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7,9] => ? = 9
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8,9] => ? = 6
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> [3,2,5,7,9,1,4,6,8] => ? = 9
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7,9] => ? = 5
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> [2,4,6,1,3,5,7,8,9] => ? = 4
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [[1,7],[2,8],[3,9],[4,10],[5],[6]]
=> [6,5,4,10,3,9,2,8,1,7] => ? = 18
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [5,10,4,9,3,8,2,7,1,6] => ? = 15
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [[1,6,10],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6,10] => ? = 14
[4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [[1,5,9],[2,6,10],[3,7],[4,8]]
=> [4,8,3,7,2,6,10,1,5,9] => ? = 16
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [[1,5,8],[2,6,9],[3,7,10],[4]]
=> [4,3,7,10,2,6,9,1,5,8] => ? = 16
[3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [[1,4,7,10],[2,5,8],[3,6,9]]
=> [3,6,9,2,5,8,1,4,7,10] => ? = 10
[3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [[1,4,7,9],[2,5,8,10],[3,6]]
=> [3,6,2,5,8,10,1,4,7,9] => ? = 12
[3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [[1,4,7,8,9,10],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8,9,10] => ? = 6
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => 6
[2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [[1,3,5,7,9,10],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7,9,10] => ? = 5
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> [[1,6,11],[2,7],[3,8],[4,9],[5,10]]
=> ? => ? = 15
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [[1,6,10],[2,7,11],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,11,1,6,10] => ? = 23
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8]]
=> [4,8,3,7,11,2,6,10,1,5,9] => ? = 19
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9]]
=> [3,6,9,2,5,8,11,1,4,7,10] => ? = 14
[2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> [[1,3,5,7,9,11],[2,4,6,8,10]]
=> ? => ? = 6
[2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ?
=> ? => ? = 5
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
=> [6,12,5,11,4,10,3,9,2,8,1,7] => ? = 21
[5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> [[1,6,11],[2,7,12],[3,8],[4,9],[5,10]]
=> ? => ? = 25
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8,12]]
=> ? => ? = 20
[4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [[1,5,9,12],[2,6,10],[3,7,11],[4,8]]
=> [4,8,3,7,11,2,6,10,1,5,9,12] => ? = 19
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? => ? = 15
[2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => 7
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
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 35%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 35%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3 = 4 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 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]
=> [4,1,2,3,5] => 3 = 4 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 4 = 5 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 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]
=> [1,2,3,4,5] => 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]
=> [6,1,2,3,4,5] => ? = 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]
=> [5,1,2,3,4,6] => ? = 5 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 6 = 7 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => ? = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 5 = 6 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 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]
=> [3,1,2,4,5,6] => ? = 3 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 4 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2 = 3 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => ? = 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]
=> [1,2,3,4,5,6] => ? = 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]
=> [7,1,2,3,4,5,6] => ? = 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]
=> [6,1,2,3,4,5,7] => ? = 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]
=> [5,1,2,3,6,4] => ? = 9 - 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]
=> [5,1,2,3,4,6,7] => ? = 5 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 8 = 9 - 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]
=> [4,1,2,5,3,6] => ? = 7 - 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]
=> [4,1,2,3,5,6,7] => ? = 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]
=> [3,4,1,2,5] => 5 = 6 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 6 = 7 - 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]
=> [3,1,4,2,5,6] => ? = 5 - 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]
=> [3,1,2,4,5,6,7] => ? = 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]
=> [2,3,4,1,5] => 3 = 4 - 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]
=> [2,3,1,4,5,6] => ? = 3 - 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]
=> [2,1,3,4,5,6,7] => ? = 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]
=> [1,2,3,4,5,6,7] => ? = 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]
=> [8,1,2,3,4,5,6,7] => ? = 8 - 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]
=> [7,1,2,3,4,5,6,8] => ? = 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]
=> [6,1,2,3,4,7,5] => ? = 11 - 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]
=> [6,1,2,3,4,5,7,8] => ? = 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]
=> [5,1,2,6,3,4] => ? = 12 - 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]
=> [5,1,2,3,6,4,7] => ? = 9 - 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]
=> [5,1,2,3,4,6,7,8] => ? = 5 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 9 = 10 - 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]
=> [4,1,5,2,3,6] => ? = 9 - 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]
=> [4,1,2,5,6,3] => ? = 10 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 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]
=> [3,4,1,2,5,6] => ? = 6 - 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]
=> [3,1,4,5,2,6] => ? = 7 - 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]
=> [2,3,4,5,1] => 4 = 5 - 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]
=> [2,3,4,1,5,6] => ? = 4 - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => ? = 3 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => ? = 2 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 1 - 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]
=> [6,1,2,3,7,4,5] => ? = 15 - 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]
=> [5,1,6,2,3,4] => ? = 14 - 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]
=> [5,1,2,6,3,4,7] => ? = 12 - 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]
=> [5,1,2,3,6,7,4] => ? = 13 - 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]
=> [4,5,1,2,3,6] => ? = 10 - 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]
=> [4,1,5,2,6,3] => ? = 14 - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 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]
=> [3,4,1,5,2,6] => ? = 9 - 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,1,2,5,6,7] => ? = 6 - 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]
=> [3,1,4,5,6,2] => ? = 9 - 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]
=> [2,3,4,5,1,6] => ? = 5 - 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => ? = 4 - 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]
=> [6,1,2,7,3,4,5] => ? = 18 - 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]
=> [5,6,1,2,3,4] => ? = 15 - 1
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => ? = 14 - 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]
=> [4,5,1,2,6,3] => ? = 16 - 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]
=> [4,1,5,6,2,3] => ? = 16 - 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]
=> [3,4,5,1,2,6] => ? = 10 - 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]
=> [3,4,1,5,6,2] => ? = 12 - 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].
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 27%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 27%
Values
[1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5]
=> [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]
=> ? = 5 - 1
[4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 5 - 1
[3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[6]
=> [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]
=> ? = 6 - 1
[5,1]
=> [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]
=> ? = 5 - 1
[4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 6 - 1
[3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 5 - 1
[3,1,1,1]
=> [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]
=> ? = 3 - 1
[2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [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 - 1
[1,1,1,1,1,1]
=> [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
[7]
=> [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]
=> ? = 7 - 1
[6,1]
=> [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]
=> ? = 6 - 1
[5,2]
=> [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]
=> ? = 9 - 1
[5,1,1]
=> [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]
=> ? = 5 - 1
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 9 - 1
[4,2,1]
=> [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]
=> ? = 7 - 1
[4,1,1,1]
=> [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]
=> ? = 4 - 1
[3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 6 - 1
[3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 7 - 1
[3,2,1,1]
=> [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]
=> ? = 5 - 1
[3,1,1,1,1]
=> [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]
=> ? = 3 - 1
[2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [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]
=> 2 = 3 - 1
[2,1,1,1,1,1]
=> [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 - 1
[1,1,1,1,1,1,1]
=> [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
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 - 1
[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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[5,3]
=> [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]
=> ? = 12 - 1
[5,2,1]
=> [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]
=> ? = 9 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 10 - 1
[4,3,1]
=> [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]
=> ? = 9 - 1
[4,2,2]
=> [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]
=> ? = 10 - 1
[3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 9 - 1
[3,3,1,1]
=> [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]
=> ? = 6 - 1
[3,2,2,1]
=> [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]
=> ? = 7 - 1
[2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2,1,1]
=> [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]
=> 3 = 4 - 1
[2,2,1,1,1,1]
=> [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]
=> 2 = 3 - 1
[2,1,1,1,1,1,1]
=> [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]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1]
=> [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
[6,3]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 15 - 1
[5,4]
=> [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]
=> ? = 14 - 1
[5,3,1]
=> [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]
=> ? = 12 - 1
[5,2,2]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 13 - 1
[4,4,1]
=> [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]
=> ? = 10 - 1
[4,3,2]
=> [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]
=> ? = 14 - 1
[3,3,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 10 - 1
[3,3,2,1]
=> [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]
=> ? = 9 - 1
[3,3,1,1,1]
=> [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]
=> ? = 6 - 1
[3,2,2,2]
=> [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]
=> ? = 9 - 1
[2,2,2,2,1]
=> [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]
=> 4 = 5 - 1
[2,2,2,1,1,1]
=> [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]
=> 3 = 4 - 1
[2,2,2,2,2]
=> [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]
=> 5 = 6 - 1
[2,2,2,2,1,1]
=> [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]
=> 4 = 5 - 1
[2,2,2,2,2,1]
=> [6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,2,2,2,2,2]
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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