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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000674
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1]
=> [2]
=> [1,0,1,0]
=> 2
[3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2
[3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[4,2,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[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
[6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 0
[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]
=> 2
[5,3]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> 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]
=> 3
[4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000297
(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
St000297: 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
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 001 => 010 => 0
[1,1]
=> 110 => 011 => 110 => 2
[3]
=> 1000 => 0001 => 0010 => 0
[2,1]
=> 1010 => 0101 => 1010 => 1
[1,1,1]
=> 1110 => 0111 => 1110 => 3
[4]
=> 10000 => 00001 => 00010 => 0
[3,1]
=> 10010 => 01001 => 10010 => 1
[2,2]
=> 1100 => 0011 => 0110 => 0
[2,1,1]
=> 10110 => 01101 => 11010 => 2
[1,1,1,1]
=> 11110 => 01111 => 11110 => 4
[5]
=> 100000 => 000001 => 000010 => 0
[4,1]
=> 100010 => 010001 => 100010 => 1
[3,2]
=> 10100 => 00101 => 01010 => 0
[3,1,1]
=> 100110 => 011001 => 110010 => 2
[2,2,1]
=> 11010 => 01011 => 10110 => 1
[2,1,1,1]
=> 101110 => 011101 => 111010 => 3
[1,1,1,1,1]
=> 111110 => 011111 => 111110 => 5
[6]
=> 1000000 => 0000001 => 0000010 => 0
[5,1]
=> 1000010 => 0100001 => 1000010 => 1
[4,2]
=> 100100 => 001001 => 010010 => 0
[4,1,1]
=> 1000110 => 0110001 => 1100010 => 2
[3,3]
=> 11000 => 00011 => 00110 => 0
[3,2,1]
=> 101010 => 010101 => 101010 => 1
[3,1,1,1]
=> 1001110 => 0111001 => 1110010 => 3
[2,2,2]
=> 11100 => 00111 => 01110 => 0
[2,2,1,1]
=> 110110 => 011011 => 110110 => 2
[2,1,1,1,1]
=> 1011110 => 0111101 => 1111010 => 4
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 1111110 => 6
[7]
=> 10000000 => 00000001 => 00000010 => 0
[6,1]
=> 10000010 => 01000001 => 10000010 => 1
[5,2]
=> 1000100 => 0010001 => 0100010 => 0
[5,1,1]
=> 10000110 => 01100001 => 11000010 => 2
[4,3]
=> 101000 => 000101 => 001010 => 0
[4,2,1]
=> 1001010 => 0101001 => 1010010 => 1
[4,1,1,1]
=> 10001110 => 01110001 => 11100010 => 3
[3,3,1]
=> 110010 => 010011 => 100110 => 1
[3,2,2]
=> 101100 => 001101 => 011010 => 0
[3,2,1,1]
=> 1010110 => 0110101 => 1101010 => 2
[3,1,1,1,1]
=> 10011110 => 01111001 => 11110010 => 4
[2,2,2,1]
=> 111010 => 010111 => 101110 => 1
[2,2,1,1,1]
=> 1101110 => 0111011 => 1110110 => 3
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 11111010 => 5
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 11111110 => 7
[7,1]
=> 100000010 => 010000001 => 100000010 => 1
[6,2]
=> 10000100 => 00100001 => 01000010 => 0
[6,1,1]
=> 100000110 => 011000001 => 110000010 => 2
[5,3]
=> 1001000 => 0001001 => 0010010 => 0
[5,2,1]
=> 10001010 => 01010001 => 10100010 => 1
[5,1,1,1]
=> 100001110 => 011100001 => 111000010 => 3
[4,4]
=> 110000 => 000011 => 000110 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1]
=> 110 => 101 => 1 = 0 + 1
[1,1]
=> [2]
=> 100 => 001 => 3 = 2 + 1
[3]
=> [1,1,1]
=> 1110 => 1101 => 1 = 0 + 1
[2,1]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,1]
=> [3]
=> 1000 => 0001 => 4 = 3 + 1
[4]
=> [1,1,1,1]
=> 11110 => 11101 => 1 = 0 + 1
[3,1]
=> [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[2,2]
=> [2,2]
=> 1100 => 1001 => 1 = 0 + 1
[2,1,1]
=> [3,1]
=> 10010 => 00101 => 3 = 2 + 1
[1,1,1,1]
=> [4]
=> 10000 => 00001 => 5 = 4 + 1
[5]
=> [1,1,1,1,1]
=> 111110 => 111101 => 1 = 0 + 1
[4,1]
=> [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
[3,2]
=> [2,2,1]
=> 11010 => 10101 => 1 = 0 + 1
[3,1,1]
=> [3,1,1]
=> 100110 => 001101 => 3 = 2 + 1
[2,2,1]
=> [3,2]
=> 10100 => 01001 => 2 = 1 + 1
[2,1,1,1]
=> [4,1]
=> 100010 => 000101 => 4 = 3 + 1
[1,1,1,1,1]
=> [5]
=> 100000 => 000001 => 6 = 5 + 1
[6]
=> [1,1,1,1,1,1]
=> 1111110 => 1111101 => 1 = 0 + 1
[5,1]
=> [2,1,1,1,1]
=> 1011110 => 0111101 => 2 = 1 + 1
[4,2]
=> [2,2,1,1]
=> 110110 => 101101 => 1 = 0 + 1
[4,1,1]
=> [3,1,1,1]
=> 1001110 => 0011101 => 3 = 2 + 1
[3,3]
=> [2,2,2]
=> 11100 => 11001 => 1 = 0 + 1
[3,2,1]
=> [3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
[3,1,1,1]
=> [4,1,1]
=> 1000110 => 0001101 => 4 = 3 + 1
[2,2,2]
=> [3,3]
=> 11000 => 10001 => 1 = 0 + 1
[2,2,1,1]
=> [4,2]
=> 100100 => 001001 => 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> 1000010 => 0000101 => 5 = 4 + 1
[1,1,1,1,1,1]
=> [6]
=> 1000000 => 0000001 => 7 = 6 + 1
[7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 11111101 => 1 = 0 + 1
[6,1]
=> [2,1,1,1,1,1]
=> 10111110 => 01111101 => 2 = 1 + 1
[5,2]
=> [2,2,1,1,1]
=> 1101110 => 1011101 => 1 = 0 + 1
[5,1,1]
=> [3,1,1,1,1]
=> 10011110 => 00111101 => 3 = 2 + 1
[4,3]
=> [2,2,2,1]
=> 111010 => 110101 => 1 = 0 + 1
[4,2,1]
=> [3,2,1,1]
=> 1010110 => 0101101 => 2 = 1 + 1
[4,1,1,1]
=> [4,1,1,1]
=> 10001110 => 00011101 => 4 = 3 + 1
[3,3,1]
=> [3,2,2]
=> 101100 => 011001 => 2 = 1 + 1
[3,2,2]
=> [3,3,1]
=> 110010 => 100101 => 1 = 0 + 1
[3,2,1,1]
=> [4,2,1]
=> 1001010 => 0010101 => 3 = 2 + 1
[3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 00001101 => 5 = 4 + 1
[2,2,2,1]
=> [4,3]
=> 101000 => 010001 => 2 = 1 + 1
[2,2,1,1,1]
=> [5,2]
=> 1000100 => 0001001 => 4 = 3 + 1
[2,1,1,1,1,1]
=> [6,1]
=> 10000010 => 00000101 => 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [7]
=> 10000000 => 00000001 => 8 = 7 + 1
[7,1]
=> [2,1,1,1,1,1,1]
=> 101111110 => 011111101 => 2 = 1 + 1
[6,2]
=> [2,2,1,1,1,1]
=> 11011110 => 10111101 => 1 = 0 + 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 100111110 => 001111101 => 3 = 2 + 1
[5,3]
=> [2,2,2,1,1]
=> 1110110 => 1101101 => 1 = 0 + 1
[5,2,1]
=> [3,2,1,1,1]
=> 10101110 => 01011101 => 2 = 1 + 1
[5,1,1,1]
=> [4,1,1,1,1]
=> 100011110 => 000111101 => 4 = 3 + 1
[4,4]
=> [2,2,2,2]
=> 111100 => 111001 => 1 = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000011
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [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 = 5 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
[4,4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000475
St000475: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[2]
=> 0
[1,1]
=> 2
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 0
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 0
[4,1,1]
=> 2
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 3
[2,2,2]
=> 0
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 0
[5,1,1]
=> 2
[4,3]
=> 0
[4,2,1]
=> 1
[4,1,1,1]
=> 3
[3,3,1]
=> 1
[3,2,2]
=> 0
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
[7,1]
=> 1
[6,2]
=> 0
[6,1,1]
=> 2
[5,3]
=> 0
[5,2,1]
=> 1
[5,1,1,1]
=> 3
[4,4]
=> 0
[6,6,1]
=> ? = 1
[5,5,3]
=> ? = 0
[5,5,2,1]
=> ? = 1
[5,4,4]
=> ? = 0
[5,4,3,1]
=> ? = 1
[4,4,4,1]
=> ? = 1
[4,4,3,2]
=> ? = 0
[4,4,3,1,1]
=> ? = 2
[4,4,2,2,1]
=> ? = 1
[4,3,3,3]
=> ? = 0
[4,3,3,2,1]
=> ? = 1
[3,3,3,3,1]
=> ? = 1
[3,3,3,2,2]
=> ? = 0
[3,3,3,2,1,1]
=> ? = 2
[3,3,2,2,2,1]
=> ? = 1
[2,2,2,2,2,2,1]
=> ? = 1
[5,5,4]
=> ? = 0
[5,5,3,1]
=> ? = 1
[5,4,4,1]
=> ? = 1
[4,4,4,2]
=> ? = 0
[4,4,4,1,1]
=> ? = 2
[4,4,3,3]
=> ? = 0
[4,4,3,2,1]
=> ? = 1
[4,3,3,3,1]
=> ? = 1
[3,3,3,3,2]
=> ? = 0
[3,3,3,3,1,1]
=> ? = 2
[3,3,3,2,2,1]
=> ? = 1
[5,5,5]
=> ? = 0
[5,5,4,1]
=> ? = 1
[4,4,4,3]
=> ? = 0
[4,4,4,2,1]
=> ? = 1
[4,4,3,3,1]
=> ? = 1
[3,3,3,3,3]
=> ? = 0
[3,3,3,3,2,1]
=> ? = 1
[5,5,5,1]
=> ? = 1
[4,4,4,4]
=> ? = 0
[4,4,4,3,1]
=> ? = 1
[3,3,3,3,3,1]
=> ? = 1
[4,4,4,4,1]
=> ? = 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001107
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 89%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 89%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[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]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[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]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[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]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 0
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 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]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 5
[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]
=> 7
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 0
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[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]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 0
[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]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> 2
[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]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4
[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]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 5
[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]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 6
[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]
=> ? = 8
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 2
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 3
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 4
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 5
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 3
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
[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]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 4
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 3
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 4
[6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
[4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 3
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 1
[3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> ? = 3
[3,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> ? = 2
[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]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 3
[6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[5,5,1,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 1
[5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 2
[4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 1
[3,3,3,1,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 3
[3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> ? = 2
[3,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[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]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 2
[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]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[5,5,2,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 1
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 2
[4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
[4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[3,3,3,2,1,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 2
[3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[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]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[5,5,3,1]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000022
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[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] => 4
[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] => 0
[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] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0
[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
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 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]
=> [2,1,3,4,5] => 3
[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] => 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]
=> [6,1,2,3,4,5] => 0
[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] => 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] => 0
[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] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[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] => 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
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[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
[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] => 4
[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] => 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]
=> [7,1,2,3,4,5,6] => 0
[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] => 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] => 0
[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] => 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 0
[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] => 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] => 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 0
[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] => 2
[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] => 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 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
[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] => 5
[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] => 7
[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] => 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] => 0
[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] => 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => 0
[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] => 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] => 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[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] => ? = 2
[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] => ? = 4
[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
[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] => ? = 5
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,1,2,3,4,7,5,8] => ? = 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,1,2,5,6,3,7] => ? = 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,1,2,3,6,4,7,8] => ? = 2
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 2
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,2,5,3,6,7,8] => ? = 3
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3,6,7] => ? = 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7,8] => ? = 4
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ? = 5
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 0
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [5,1,2,3,6,7,4,8] => ? = 1
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,4,5,6,2,7] => ? = 1
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,1,2,6,7,3,5] => ? = 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,2,5,6,3,7,8] => ? = 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5,8] => ? = 1
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,1,5,6,2,4,7] => ? = 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,2,6,7,8] => ? = 3
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,1,2,6,7,3,4] => ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [5,1,2,6,3,4,7,8] => ? = 2
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,4,5,1,3,6,7] => ? = 2
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7,8] => ? = 4
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [4,1,5,6,2,3,7] => ? = 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,3,6,7,8] => ? = 3
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,1,2,6,7] => ? = 2
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 0
[6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [4,1,2,5,6,7,3,8] => ? = 1
[5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,1,4,6,7,2,5] => ? = 0
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ? = 2
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [4,1,2,6,7,3,5,8] => ? = 1
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,3,5,6,1,4,7] => ? = 1
[4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,1,5,6,7,2,4] => ? = 0
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> [3,1,5,6,2,4,7,8] => ? = 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [5,1,2,6,7,3,4,8] => ? = 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,1,2,3,6,7] => ? = 2
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,4,5,6,1,3,7] => ? = 1
[3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [2,4,5,1,3,6,7,8] => ? = 3
[3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,1,5,6,7,2,3] => ? = 0
[3,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [4,1,5,6,2,3,7,8] => ? = 2
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,6,1,2,7] => ? = 1
[2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,5,1,2,6,7,8] => ? = 3
[6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,4,5,6,7,2,8] => ? = 1
[5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,1,6,7,2,4,5] => ? = 0
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
=> [3,1,4,6,7,2,5,8] => ? = 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,1,3,4,7] => ? = 1
[4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,3,5,6,1,4,7,8] => ? = 2
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,1,6,2,7,3,4] => ? = 0
[4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [5,1,6,2,3,4,7,8] => ? = 2
Description
The number of fixed points of a permutation.
Matching statistic: St000873
(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
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 89%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 89%
Values
[2]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[1,1]
=> [2]
=> [1,0,1,0]
=> [1,2] => 2
[3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0
[3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
[5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0
[4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 2
[3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3
[2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 4
[1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 6
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => 1
[5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => 2
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 0
[4,2,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => 3
[3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 0
[3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => 4
[2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 3
[2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => 5
[1,1,1,1,1,1,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 7
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? = 1
[6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ? = 0
[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]
=> [1,2,4,5,6,7,8,3] => ? = 2
[5,3]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => 0
[5,2,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => 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]
=> [1,2,3,5,6,7,8,4] => ? = 3
[4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 0
[4,3,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => 1
[4,2,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,2,5,6,1] => 0
[4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ? = 4
[3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 0
[3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 2
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ? = 5
[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]
=> [1,2,3,4,5,6,8,7] => ? = 6
[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]
=> [1,2,3,4,5,6,7,8] => ? = 8
[6,3]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ? = 0
[6,2,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => ? = 1
[5,3,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 1
[5,2,2]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,1] => ? = 0
[5,2,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => ? = 2
[4,2,1,1,1]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => ? = 3
[3,2,1,1,1,1]
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => ? = 4
[2,2,1,1,1,1,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ? = 5
[6,4]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 0
[6,3,1]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,4,3,6,7,8,2] => ? = 1
[5,4,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 1
[5,3,2]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? = 0
[5,3,1,1]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => ? = 2
[5,2,2,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,3,6,7,8,2] => ? = 1
[4,3,1,1,1]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => ? = 3
[4,2,2,2]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => ? = 0
[4,2,2,1,1]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,4,7,8,3] => ? = 2
[3,3,1,1,1,1]
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => ? = 4
[3,2,2,1,1,1]
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,5,8,4] => ? = 3
[2,2,2,1,1,1,1]
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,6,5] => ? = 4
[6,5]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 0
[6,4,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,5,6,4,3,7,8,2] => ? = 1
[5,5,1]
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 1
[5,4,2]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => ? = 0
[5,4,1,1]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,5,4,8,3] => ? = 2
[5,3,3]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 0
[5,3,2,1]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,5,3,7,8,2] => ? = 1
[4,4,2,1]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 1
[4,4,1,1,1]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,6,5,4] => ? = 3
[4,3,3,1]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 1
[4,3,2,2]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => ? = 0
[4,3,2,1,1]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,6,4,8,3] => ? = 2
[4,2,2,2,1]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,6,3,7,8,2] => ? = 1
[3,3,2,1,1,1]
=> [6,3,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,7,5,4] => ? = 3
[3,2,2,2,2]
=> [5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,2,7,1] => ? = 0
[3,2,2,2,1,1]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,2,5,6,7,4,8,3] => ? = 2
[2,2,2,2,1,1,1]
=> [7,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,6,7,8,5,4] => ? = 3
[6,6]
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[6,5,1]
=> [3,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,5,6,7,4,3,8,2] => ? = 1
[5,5,2]
=> [3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => ? = 0
[5,5,1,1]
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,5,4,3] => ? = 2
Description
The aix statistic of a permutation.
According to [1], this statistic on finite strings $\pi$ of integers is given as follows: let $m$ be the leftmost occurrence of the minimal entry and let $\pi = \alpha\ m\ \beta$. Then
$$
\operatorname{aix}\pi = \begin{cases}
\operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\
1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\
0 & \text{ if } \beta = \emptyset
\end{cases}\ .
$$
Matching statistic: St000895
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 78%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 78%
Values
[2]
=> [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,1]
=> [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 0
[2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
[1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 1
[4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2
[3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 0
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4
[1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 6
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[4,2,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 4
[2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 3
[2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5
[1,1,1,1,1,1,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 7
[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]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 1
[6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 2
[5,3]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 3
[4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[4,3,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 1
[4,2,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
[4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 4
[3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 2
[3,2,2,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 1
[3,2,1,1,1]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 5
[2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 2
[2,2,1,1,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4
[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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6
[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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 8
[6,3]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[6,2,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 1
[5,4]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
[5,3,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[5,2,2]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[5,2,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 2
[4,4,1]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 1
[4,3,2]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
[4,3,1,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[4,2,2,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[4,2,1,1,1]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 3
[3,3,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[3,3,2,1]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 1
[3,3,1,1,1]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[3,2,2,1,1]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[3,2,1,1,1,1]
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 4
[2,2,2,1,1,1]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,2,1,1,1,1,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? = 5
[6,4]
=> [2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,-1,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[6,3,1]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 1
[5,4,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,-1,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[5,3,2]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[5,3,1,1]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 2
[5,2,2,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 1
[4,4,1,1]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[4,3,1,1,1]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 3
[4,2,2,2]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
[4,2,2,1,1]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 2
[3,3,2,1,1]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[3,3,1,1,1,1]
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? = 4
[3,2,2,2,1]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[3,2,2,1,1,1]
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 3
[2,2,2,2,1,1]
=> [6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,2,2,1,1,1,1]
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? = 4
[6,5]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,-1,1,0,0],[0,1,0,0,-1,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 0
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St000241
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 78%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 78%
Values
[2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? = 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => 0
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 7
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => ? = 0
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => ? = 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => ? = 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ? = 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => 2
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => ? = 3
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ? = 5
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? = 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? = 6
[1,1,1,1,1,1,1,1]
=> [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
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => ? = 0
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,5,8,6] => ? = 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => ? = 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 0
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,4,7,8,5] => ? = 2
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => 0
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,5,2,6,7,3] => ? = 2
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => ? = 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,3,6,7,8,4] => ? = 3
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? = 3
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,8,3] => ? = 4
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? = 3
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 5
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => ? = 0
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,4,8,5] => ? = 1
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,2,7,3] => ? = 1
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => ? = 0
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,3,7,8,4] => ? = 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,4,5,8,6] => ? = 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,1,6,7,2] => ? = 2
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,2,3,7,5] => ? = 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,2,6,7,8,3] => ? = 3
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => ? = 0
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,3,4,7,8,5] => ? = 2
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,1,2,6,7,4] => ? = 2
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,8,2] => ? = 4
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,3,7,4] => ? = 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,2,3,6,7,8,4] => ? = 3
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,1,2,6,7,3] => ? = 2
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 4
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,2,3] => ? = 0
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000248The number of anti-singletons of a set partition. St000247The number of singleton blocks of a set partition. St000215The number of adjacencies of a permutation, zero appended. St001459The number of zero columns in the nullspace of a graph.
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