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Your data matches 113 different statistics following compositions of up to 3 maps.
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Matching statistic: St001389
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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)
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
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 7
[8]
=> []
=> 1
[7,1]
=> [1]
=> 2
[6,2]
=> [2]
=> 3
[6,1,1]
=> [1,1]
=> 3
[5,3]
=> [3]
=> 4
[5,2,1]
=> [2,1]
=> 5
Description
The number of partitions contained in the given partition.
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
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 10111111 => 7
[8]
=> 100000000 => 000000001 => 100000000 => 1
[7,1]
=> 100000010 => 010000001 => 101000000 => 2
[6,2]
=> 10000100 => 00100001 => 10010000 => 3
[6,1,1]
=> 100000110 => 011000001 => 101100000 => 3
[5,3]
=> 1001000 => 0001001 => 1000100 => 4
[5,2,1]
=> 10001010 => 01010001 => 10101000 => 5
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: St000110
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●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: 94% ●values known / values provided: 94%●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
[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]
=> [4,1,2,5,3,6,7] => ? ∊ {3,4,5,7}
[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]
=> [4,1,2,3,5,6,7,8] => ? ∊ {3,4,5,7}
[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]
=> [3,1,4,2,5,6,7] => ? ∊ {3,4,5,7}
[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]
=> [3,1,2,4,5,6,7,8] => ? ∊ {3,4,5,7}
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: 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: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 92%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 92%
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
[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
[8]
=> []
=> []
=> ? = 1
[7,1]
=> [1]
=> [1,0,1,0]
=> 2
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 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
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[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
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[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
Description
The number of Dyck paths that are weakly above a Dyck path.
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: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 92%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 92%
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
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7
[8]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[7,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[6,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[6,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(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
[5,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[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
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[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
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: St001464
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
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: 83% ●values known / values provided: 85%●distinct values known / distinct values provided: 83%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 85%●distinct values known / distinct values provided: 83%
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
[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]
=> [2,3,4,5,6,7,1] => ? = 7
[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]
=> [1,2,3,4,5,6,7,8] => ? ∊ {1,2,3,4,5,6,7,8,11}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => ? ∊ {1,2,3,4,5,6,7,8,11}
[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
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,7,8,6] => ? ∊ {1,2,3,4,5,6,7,8,11}
[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
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,5] => ? ∊ {1,2,3,4,5,6,7,8,11}
[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
[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]
=> [1,2,3,5,6,7,8,4] => ? ∊ {1,2,3,4,5,6,7,8,11}
[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]
=> [1,2,4,5,6,7,8,3] => ? ∊ {1,2,3,4,5,6,7,8,11}
[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]
=> [3,1,4,5,6,7,2] => ? ∊ {1,2,3,4,5,6,7,8,11}
[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]
=> [1,3,4,5,6,7,8,2] => ? ∊ {1,2,3,4,5,6,7,8,11}
[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]
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,2,3,4,5,6,7,8,11}
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St000883
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
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] => ? ∊ {4,5,6,7,7,9,9}
[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] => ? ∊ {4,5,6,7,7,9,9}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ? ∊ {4,5,6,7,7,9,9}
[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] => ? ∊ {4,5,6,7,7,9,9}
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? ∊ {4,5,6,7,7,9,9}
[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] => ? ∊ {4,5,6,7,7,9,9}
[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] => ? ∊ {4,5,6,7,7,9,9}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,5,7,8],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => 4
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[1,4,6,7,8],[2,5],[3]]
=> [3,2,5,1,4,6,7,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => 3
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[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
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001232
(load all 197 compositions to match this statistic)
(load all 197 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 61%●distinct values known / distinct values provided: 58%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 61%●distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 5 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {4,6,7} - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {4,6,7} - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {4,6,7} - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 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]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {4,7,7,9,9} - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {4,7,7,9,9} - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {4,7,7,9,9} - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {4,7,7,9,9} - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {4,7,7,9,9} - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 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]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 6 = 7 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,1,1,1,1,1]
=> [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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,6,7,8,9,9,9,10,10,11,12} - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000369
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 55%●distinct values known / distinct values provided: 42%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 55%●distinct values known / distinct values provided: 42%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 3 = 4 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? ∊ {5,5} - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {5,5} - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? ∊ {5,6,6,7} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? ∊ {5,6,6,7} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> 3 = 4 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {5,6,6,7} - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {5,6,6,7} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> ? ∊ {5,6,6,7,7,7,9,9} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0]
=> ? ∊ {5,6,6,7,7,7,9,9} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? ∊ {5,6,6,7,7,7,9,9} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {5,6,6,7,7,7,9,9} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {5,6,6,7,7,7,9,9} - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {5,6,6,7,7,7,9,9} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {5,6,6,7,7,7,9,9} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {5,6,6,7,7,7,9,9} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {4,5,5,6,6,7,7,7,8,9,9,9,10,10,11,12} - 1
Description
The dinv deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$
In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$.
See also [[St000376]] for the bounce deficit.
The following 103 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001684The reduced word complexity of a permutation. St000004The major index of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000797The stat`` of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000133The "bounce" of a permutation. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001330The hat guessing number of a graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000456The monochromatic index of a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000120The number of left tunnels of a Dyck path. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000422The energy of a graph, if it is integral. St000653The last descent of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001497The position of the largest weak excedence of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001821The sorting index of a signed 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 $K[x]/(x^n)$. St000766The number of inversions of an integer composition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001596The number of two-by-two squares inside a skew partition. St000006The dinv of a Dyck path. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000691The number of changes of a binary word. St001267The length of the Lyndon factorization of the binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St000031The number of cycles in the cycle decomposition of a permutation. St000141The maximum drop size of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001096The size of the overlap set of a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St000662The staircase size of the code of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000056The decomposition (or block) number of a permutation. St000091The descent variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000314The number of left-to-right-maxima of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000991The number of right-to-left minima of a permutation. St001060The distinguishing index of a graph. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001461The number of topologically connected components of the chord diagram of a permutation. St001645The pebbling number of a connected graph. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000173The segment statistic of a semistandard tableau. St000234The number of global ascents of a permutation. St000317The cycle descent number of a permutation. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000491The number of inversions of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000650The number of 3-rises of a permutation. St001403The number of vertical separators in a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001843The Z-index of a set partition. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000834The number of right outer peaks of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001087The number of occurrences of the vincular pattern |12-3 in a permutation.
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