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Your data matches 70 different statistics following compositions of up to 3 maps.
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Matching statistic: St000475
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 = 2 - 1
[2]
=> 0 = 1 - 1
[1,1]
=> 2 = 3 - 1
[3]
=> 0 = 1 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 3 = 4 - 1
[4]
=> 0 = 1 - 1
[3,1]
=> 1 = 2 - 1
[2,2]
=> 0 = 1 - 1
[2,1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> 4 = 5 - 1
[5]
=> 0 = 1 - 1
[4,1]
=> 1 = 2 - 1
[3,2]
=> 0 = 1 - 1
[3,1,1]
=> 2 = 3 - 1
[2,2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> 5 = 6 - 1
[6]
=> 0 = 1 - 1
[5,1]
=> 1 = 2 - 1
[4,2]
=> 0 = 1 - 1
[4,1,1]
=> 2 = 3 - 1
[3,3]
=> 0 = 1 - 1
[3,2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> 3 = 4 - 1
[2,2,2]
=> 0 = 1 - 1
[2,2,1,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> 6 = 7 - 1
[7]
=> 0 = 1 - 1
[6,1]
=> 1 = 2 - 1
[5,2]
=> 0 = 1 - 1
[5,1,1]
=> 2 = 3 - 1
[4,3]
=> 0 = 1 - 1
[4,2,1]
=> 1 = 2 - 1
[4,1,1,1]
=> 3 = 4 - 1
[3,3,1]
=> 1 = 2 - 1
[3,2,2]
=> 0 = 1 - 1
[3,2,1,1]
=> 2 = 3 - 1
[3,1,1,1,1]
=> 4 = 5 - 1
[2,2,2,1]
=> 1 = 2 - 1
[2,2,1,1,1]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> 7 = 8 - 1
[8]
=> 0 = 1 - 1
[7,1]
=> 1 = 2 - 1
[6,2]
=> 0 = 1 - 1
[6,1,1]
=> 2 = 3 - 1
[5,3]
=> 0 = 1 - 1
[5,2,1]
=> 1 = 2 - 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000326
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
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%
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
[1]
=> 10 => 01 => 2
[2]
=> 100 => 001 => 3
[1,1]
=> 110 => 101 => 1
[3]
=> 1000 => 0001 => 4
[2,1]
=> 1010 => 0101 => 2
[1,1,1]
=> 1110 => 1101 => 1
[4]
=> 10000 => 00001 => 5
[3,1]
=> 10010 => 00101 => 3
[2,2]
=> 1100 => 1001 => 1
[2,1,1]
=> 10110 => 01101 => 2
[1,1,1,1]
=> 11110 => 11101 => 1
[5]
=> 100000 => 000001 => 6
[4,1]
=> 100010 => 000101 => 4
[3,2]
=> 10100 => 01001 => 2
[3,1,1]
=> 100110 => 001101 => 3
[2,2,1]
=> 11010 => 10101 => 1
[2,1,1,1]
=> 101110 => 011101 => 2
[1,1,1,1,1]
=> 111110 => 111101 => 1
[6]
=> 1000000 => 0000001 => 7
[5,1]
=> 1000010 => 0000101 => 5
[4,2]
=> 100100 => 001001 => 3
[4,1,1]
=> 1000110 => 0001101 => 4
[3,3]
=> 11000 => 10001 => 1
[3,2,1]
=> 101010 => 010101 => 2
[3,1,1,1]
=> 1001110 => 0011101 => 3
[2,2,2]
=> 11100 => 11001 => 1
[2,2,1,1]
=> 110110 => 101101 => 1
[2,1,1,1,1]
=> 1011110 => 0111101 => 2
[1,1,1,1,1,1]
=> 1111110 => 1111101 => 1
[7]
=> 10000000 => 00000001 => 8
[6,1]
=> 10000010 => 00000101 => 6
[5,2]
=> 1000100 => 0001001 => 4
[5,1,1]
=> 10000110 => 00001101 => 5
[4,3]
=> 101000 => 010001 => 2
[4,2,1]
=> 1001010 => 0010101 => 3
[4,1,1,1]
=> 10001110 => 00011101 => 4
[3,3,1]
=> 110010 => 100101 => 1
[3,2,2]
=> 101100 => 011001 => 2
[3,2,1,1]
=> 1010110 => 0101101 => 2
[3,1,1,1,1]
=> 10011110 => 00111101 => 3
[2,2,2,1]
=> 111010 => 110101 => 1
[2,2,1,1,1]
=> 1101110 => 1011101 => 1
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 2
[1,1,1,1,1,1,1]
=> 11111110 => 11111101 => 1
[8]
=> 100000000 => 000000001 => 9
[7,1]
=> 100000010 => 000000101 => 7
[6,2]
=> 10000100 => 00001001 => 5
[6,1,1]
=> 100000110 => 000001101 => 6
[5,3]
=> 1001000 => 0010001 => 3
[5,2,1]
=> 10001010 => 00010101 => 4
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: St000877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 => 1 = 2 - 1
[2]
=> 0 => 1 => 0 = 1 - 1
[1,1]
=> 11 => 00 => 2 = 3 - 1
[3]
=> 1 => 0 => 1 = 2 - 1
[2,1]
=> 01 => 10 => 0 = 1 - 1
[1,1,1]
=> 111 => 000 => 3 = 4 - 1
[4]
=> 0 => 1 => 0 = 1 - 1
[3,1]
=> 11 => 00 => 2 = 3 - 1
[2,2]
=> 00 => 11 => 0 = 1 - 1
[2,1,1]
=> 011 => 100 => 1 = 2 - 1
[1,1,1,1]
=> 1111 => 0000 => 4 = 5 - 1
[5]
=> 1 => 0 => 1 = 2 - 1
[4,1]
=> 01 => 10 => 0 = 1 - 1
[3,2]
=> 10 => 01 => 1 = 2 - 1
[3,1,1]
=> 111 => 000 => 3 = 4 - 1
[2,2,1]
=> 001 => 110 => 0 = 1 - 1
[2,1,1,1]
=> 0111 => 1000 => 2 = 3 - 1
[1,1,1,1,1]
=> 11111 => 00000 => 5 = 6 - 1
[6]
=> 0 => 1 => 0 = 1 - 1
[5,1]
=> 11 => 00 => 2 = 3 - 1
[4,2]
=> 00 => 11 => 0 = 1 - 1
[4,1,1]
=> 011 => 100 => 1 = 2 - 1
[3,3]
=> 11 => 00 => 2 = 3 - 1
[3,2,1]
=> 101 => 010 => 1 = 2 - 1
[3,1,1,1]
=> 1111 => 0000 => 4 = 5 - 1
[2,2,2]
=> 000 => 111 => 0 = 1 - 1
[2,2,1,1]
=> 0011 => 1100 => 0 = 1 - 1
[2,1,1,1,1]
=> 01111 => 10000 => 3 = 4 - 1
[1,1,1,1,1,1]
=> 111111 => 000000 => 6 = 7 - 1
[7]
=> 1 => 0 => 1 = 2 - 1
[6,1]
=> 01 => 10 => 0 = 1 - 1
[5,2]
=> 10 => 01 => 1 = 2 - 1
[5,1,1]
=> 111 => 000 => 3 = 4 - 1
[4,3]
=> 01 => 10 => 0 = 1 - 1
[4,2,1]
=> 001 => 110 => 0 = 1 - 1
[4,1,1,1]
=> 0111 => 1000 => 2 = 3 - 1
[3,3,1]
=> 111 => 000 => 3 = 4 - 1
[3,2,2]
=> 100 => 011 => 1 = 2 - 1
[3,2,1,1]
=> 1011 => 0100 => 2 = 3 - 1
[3,1,1,1,1]
=> 11111 => 00000 => 5 = 6 - 1
[2,2,2,1]
=> 0001 => 1110 => 0 = 1 - 1
[2,2,1,1,1]
=> 00111 => 11000 => 1 = 2 - 1
[2,1,1,1,1,1]
=> 011111 => 100000 => 4 = 5 - 1
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 7 = 8 - 1
[8]
=> 0 => 1 => 0 = 1 - 1
[7,1]
=> 11 => 00 => 2 = 3 - 1
[6,2]
=> 00 => 11 => 0 = 1 - 1
[6,1,1]
=> 011 => 100 => 1 = 2 - 1
[5,3]
=> 11 => 00 => 2 = 3 - 1
[5,2,1]
=> 101 => 010 => 1 = 2 - 1
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000297
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Mp00278: Binary words —rowmotion⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Values
[1]
=> 1 => 1 => 1 => 1 = 2 - 1
[2]
=> 0 => 0 => 0 => 0 = 1 - 1
[1,1]
=> 11 => 11 => 11 => 2 = 3 - 1
[3]
=> 1 => 1 => 1 => 1 = 2 - 1
[2,1]
=> 01 => 10 => 01 => 0 = 1 - 1
[1,1,1]
=> 111 => 111 => 111 => 3 = 4 - 1
[4]
=> 0 => 0 => 0 => 0 = 1 - 1
[3,1]
=> 11 => 11 => 11 => 2 = 3 - 1
[2,2]
=> 00 => 00 => 00 => 0 = 1 - 1
[2,1,1]
=> 011 => 101 => 101 => 1 = 2 - 1
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4 = 5 - 1
[5]
=> 1 => 1 => 1 => 1 = 2 - 1
[4,1]
=> 01 => 10 => 01 => 0 = 1 - 1
[3,2]
=> 10 => 01 => 10 => 1 = 2 - 1
[3,1,1]
=> 111 => 111 => 111 => 3 = 4 - 1
[2,2,1]
=> 001 => 010 => 010 => 0 = 1 - 1
[2,1,1,1]
=> 0111 => 1011 => 1101 => 2 = 3 - 1
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5 = 6 - 1
[6]
=> 0 => 0 => 0 => 0 = 1 - 1
[5,1]
=> 11 => 11 => 11 => 2 = 3 - 1
[4,2]
=> 00 => 00 => 00 => 0 = 1 - 1
[4,1,1]
=> 011 => 101 => 101 => 1 = 2 - 1
[3,3]
=> 11 => 11 => 11 => 2 = 3 - 1
[3,2,1]
=> 101 => 110 => 011 => 0 = 1 - 1
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4 = 5 - 1
[2,2,2]
=> 000 => 000 => 000 => 0 = 1 - 1
[2,2,1,1]
=> 0011 => 0101 => 1010 => 1 = 2 - 1
[2,1,1,1,1]
=> 01111 => 10111 => 11101 => 3 = 4 - 1
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6 = 7 - 1
[7]
=> 1 => 1 => 1 => 1 = 2 - 1
[6,1]
=> 01 => 10 => 01 => 0 = 1 - 1
[5,2]
=> 10 => 01 => 10 => 1 = 2 - 1
[5,1,1]
=> 111 => 111 => 111 => 3 = 4 - 1
[4,3]
=> 01 => 10 => 01 => 0 = 1 - 1
[4,2,1]
=> 001 => 010 => 010 => 0 = 1 - 1
[4,1,1,1]
=> 0111 => 1011 => 1101 => 2 = 3 - 1
[3,3,1]
=> 111 => 111 => 111 => 3 = 4 - 1
[3,2,2]
=> 100 => 001 => 100 => 1 = 2 - 1
[3,2,1,1]
=> 1011 => 1101 => 1011 => 1 = 2 - 1
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5 = 6 - 1
[2,2,2,1]
=> 0001 => 0010 => 0100 => 0 = 1 - 1
[2,2,1,1,1]
=> 00111 => 01011 => 11010 => 2 = 3 - 1
[2,1,1,1,1,1]
=> 011111 => 101111 => 111101 => 4 = 5 - 1
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7 = 8 - 1
[8]
=> 0 => 0 => 0 => 0 = 1 - 1
[7,1]
=> 11 => 11 => 11 => 2 = 3 - 1
[6,2]
=> 00 => 00 => 00 => 0 = 1 - 1
[6,1,1]
=> 011 => 101 => 101 => 1 = 2 - 1
[5,3]
=> 11 => 11 => 11 => 2 = 3 - 1
[5,2,1]
=> 101 => 110 => 011 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? = 11 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000011
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[6]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[7]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [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
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 9
[7,1]
=> [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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,3}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,3}
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: St001462
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 92%●distinct values known / distinct values provided: 73%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 92%●distinct values known / distinct values provided: 73%
Values
[1]
=> []
=> []
=> []
=> ? = 2
[2]
=> []
=> []
=> []
=> ? = 3
[1,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[3]
=> []
=> []
=> []
=> ? = 4
[2,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[4]
=> []
=> []
=> []
=> ? = 5
[3,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[5]
=> []
=> []
=> []
=> ? = 6
[4,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[3,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[2,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[6]
=> []
=> []
=> []
=> ? = 7
[5,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[4,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[3,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[2,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5
[7]
=> []
=> []
=> []
=> ? = 8
[6,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[5,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[4,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 6
[8]
=> []
=> []
=> []
=> ? = 9
[7,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[6,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[5,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[5,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[4,4]
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[9]
=> []
=> []
=> []
=> ? = 10
[10]
=> []
=> []
=> []
=> ? ∊ {9,11}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [[1,2,3,4,5,6,7,8,9]]
=> ? ∊ {9,11}
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St001050
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 88%●distinct values known / distinct values provided: 64%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 64% ●values known / values provided: 88%●distinct values known / distinct values provided: 64%
Values
[1]
=> []
=> []
=> {}
=> ? = 2
[2]
=> []
=> []
=> {}
=> ? = 3
[1,1]
=> [1]
=> [[1]]
=> {{1}}
=> 1
[3]
=> []
=> []
=> {}
=> ? = 4
[2,1]
=> [1]
=> [[1]]
=> {{1}}
=> 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[4]
=> []
=> []
=> {}
=> ? = 5
[3,1]
=> [1]
=> [[1]]
=> {{1}}
=> 1
[2,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[5]
=> []
=> []
=> {}
=> ? = 6
[4,1]
=> [1]
=> [[1]]
=> {{1}}
=> 1
[3,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4
[6]
=> []
=> []
=> {}
=> ? = 7
[5,1]
=> [1]
=> [[1]]
=> {{1}}
=> 1
[4,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[3,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 5
[7]
=> []
=> []
=> {}
=> ? = 8
[6,1]
=> [1]
=> [[1]]
=> {{1}}
=> 1
[5,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[4,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 6
[8]
=> []
=> []
=> {}
=> ? = 9
[7,1]
=> [1]
=> [[1]]
=> {{1}}
=> 1
[6,2]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[5,3]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[4,4]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3
[9]
=> []
=> []
=> {}
=> ? ∊ {8,10}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {8,10}
[10]
=> []
=> []
=> {}
=> ? ∊ {3,5,7,8,9,11}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? ∊ {3,5,7,8,9,11}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? ∊ {3,5,7,8,9,11}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {3,5,7,8,9,11}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {3,5,7,8,9,11}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? ∊ {3,5,7,8,9,11}
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000382
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 91%
Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 91%
Values
[1]
=> 10 => 00 => [2] => 2
[2]
=> 100 => 000 => [3] => 3
[1,1]
=> 110 => 010 => [1,1,1] => 1
[3]
=> 1000 => 0000 => [4] => 4
[2,1]
=> 1010 => 0010 => [2,1,1] => 2
[1,1,1]
=> 1110 => 0110 => [1,2,1] => 1
[4]
=> 10000 => 00000 => [5] => 5
[3,1]
=> 10010 => 00010 => [3,1,1] => 3
[2,2]
=> 1100 => 0100 => [1,1,2] => 1
[2,1,1]
=> 10110 => 00110 => [2,2,1] => 2
[1,1,1,1]
=> 11110 => 01110 => [1,3,1] => 1
[5]
=> 100000 => 000000 => [6] => 6
[4,1]
=> 100010 => 000010 => [4,1,1] => 4
[3,2]
=> 10100 => 00100 => [2,1,2] => 2
[3,1,1]
=> 100110 => 000110 => [3,2,1] => 3
[2,2,1]
=> 11010 => 01010 => [1,1,1,1,1] => 1
[2,1,1,1]
=> 101110 => 001110 => [2,3,1] => 2
[1,1,1,1,1]
=> 111110 => 011110 => [1,4,1] => 1
[6]
=> 1000000 => 0000000 => [7] => 7
[5,1]
=> 1000010 => 0000010 => [5,1,1] => 5
[4,2]
=> 100100 => 000100 => [3,1,2] => 3
[4,1,1]
=> 1000110 => 0000110 => [4,2,1] => 4
[3,3]
=> 11000 => 01000 => [1,1,3] => 1
[3,2,1]
=> 101010 => 001010 => [2,1,1,1,1] => 2
[3,1,1,1]
=> 1001110 => 0001110 => [3,3,1] => 3
[2,2,2]
=> 11100 => 01100 => [1,2,2] => 1
[2,2,1,1]
=> 110110 => 010110 => [1,1,1,2,1] => 1
[2,1,1,1,1]
=> 1011110 => 0011110 => [2,4,1] => 2
[1,1,1,1,1,1]
=> 1111110 => 0111110 => [1,5,1] => 1
[7]
=> 10000000 => 00000000 => [8] => 8
[6,1]
=> 10000010 => 00000010 => [6,1,1] => 6
[5,2]
=> 1000100 => 0000100 => [4,1,2] => 4
[5,1,1]
=> 10000110 => 00000110 => [5,2,1] => 5
[4,3]
=> 101000 => 001000 => [2,1,3] => 2
[4,2,1]
=> 1001010 => 0001010 => [3,1,1,1,1] => 3
[4,1,1,1]
=> 10001110 => 00001110 => [4,3,1] => 4
[3,3,1]
=> 110010 => 010010 => [1,1,2,1,1] => 1
[3,2,2]
=> 101100 => 001100 => [2,2,2] => 2
[3,2,1,1]
=> 1010110 => 0010110 => [2,1,1,2,1] => 2
[3,1,1,1,1]
=> 10011110 => 00011110 => [3,4,1] => 3
[2,2,2,1]
=> 111010 => 011010 => [1,2,1,1,1] => 1
[2,2,1,1,1]
=> 1101110 => 0101110 => [1,1,1,3,1] => 1
[2,1,1,1,1,1]
=> 10111110 => 00111110 => [2,5,1] => 2
[1,1,1,1,1,1,1]
=> 11111110 => 01111110 => [1,6,1] => 1
[8]
=> 100000000 => 000000000 => [9] => 9
[7,1]
=> 100000010 => 000000010 => [7,1,1] => 7
[6,2]
=> 10000100 => 00000100 => [5,1,2] => 5
[6,1,1]
=> 100000110 => 000000110 => [6,2,1] => 6
[5,3]
=> 1001000 => 0001000 => [3,1,3] => 3
[5,2,1]
=> 10001010 => 00001010 => [4,1,1,1,1] => 4
[3,1,1,1,1,1,1]
=> 1001111110 => 0001111110 => [3,6,1] => ? ∊ {2,3}
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0011111110 => [2,7,1] => ? ∊ {2,3}
[10]
=> 10000000000 => 00000000000 => [11] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[9,1]
=> 10000000010 => 00000000010 => [9,1,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[8,2]
=> 1000000100 => 0000000100 => [7,1,2] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[8,1,1]
=> 10000000110 => 00000000110 => [8,2,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[7,1,1,1]
=> 10000001110 => 00000001110 => [7,3,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[6,2,1,1]
=> 1000010110 => 0000010110 => [5,1,1,2,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[6,1,1,1,1]
=> 10000011110 => 00000011110 => [6,4,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[5,2,1,1,1]
=> 1000101110 => 0000101110 => [4,1,1,3,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[5,1,1,1,1,1]
=> 10000111110 => 00000111110 => [5,5,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[4,2,1,1,1,1]
=> 1001011110 => 0001011110 => [3,1,1,4,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[4,1,1,1,1,1,1]
=> 10001111110 => 00001111110 => [4,6,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[3,2,1,1,1,1,1]
=> 1010111110 => 0010111110 => [2,1,1,5,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[3,1,1,1,1,1,1,1]
=> 10011111110 => 00011111110 => [3,7,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[2,2,1,1,1,1,1,1]
=> 1101111110 => 0101111110 => [1,1,1,6,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 00111111110 => [2,8,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 01111111110 => [1,9,1] => ? ∊ {1,1,2,2,3,3,4,4,5,5,6,7,7,8,9,11}
Description
The first part of an integer composition.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 91%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 91%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [2,1] => 2
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,3,2,1] => 4
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => 5
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,4,2,1] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [6,5,4,3,2,1] => 6
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,5,1,3,4] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [2,1,3,4,5,6] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [7,6,5,4,3,2,1] => 7
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [5,6,4,3,2,1] => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,5,4,2,1] => 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [4,3,2,1,5] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,4,1,3] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,1,3,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [2,6,1,3,4,5] => 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [2,1,3,4,5,6,7] => 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [8,7,6,5,4,3,2,1] => 8
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [6,7,5,4,3,2,1] => ? = 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [5,4,6,3,2,1] => 4
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [4,6,5,3,2,1] => 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,5,4,3,1] => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [2,6,5,1,3,4] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [2,5,1,3,4,6] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [2,7,1,3,4,5,6] => 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8] => 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [9,8,7,6,5,4,3,2,1] => 9
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [7,8,6,5,4,3,2,1] => ? ∊ {3,5,6,7}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [6,5,7,4,3,2,1] => ? ∊ {3,5,6,7}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [5,7,6,4,3,2,1] => ? ∊ {3,5,6,7}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [5,4,3,6,2,1] => 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [4,5,6,3,2,1] => 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [3,6,5,4,2,1] => 5
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [2,7,6,1,3,4,5] => ? ∊ {3,5,6,7}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,6,5,4,3,2,1,9] => [8,9,7,6,5,4,3,2,1] => ? ∊ {3,4,4,5,6,7,8}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,6,4,3,2,1,8] => [6,8,7,5,4,3,2,1] => ? ∊ {3,4,4,5,6,7,8}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => [6,5,4,7,3,2,1] => ? ∊ {3,4,4,5,6,7,8}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => [5,6,7,4,3,2,1] => ? ∊ {3,4,4,5,6,7,8}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => [4,7,6,5,3,2,1] => ? ∊ {3,4,4,5,6,7,8}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [2,7,6,5,1,3,4] => ? ∊ {3,4,4,5,6,7,8}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => [2,8,7,1,3,4,5,6] => ? ∊ {3,4,4,5,6,7,8}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [11,10,9,8,7,6,5,4,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,7,6,5,4,3,2,1,10] => [9,10,8,7,6,5,4,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [8,7,9,6,5,4,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,7,5,4,3,2,1,9] => [7,9,8,6,5,4,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,5,4,7,3,2,1,8] => [7,6,5,8,4,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,6,7,4,3,2,1,8] => [6,7,8,5,4,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,7,6,5,3,2,1,8] => [5,8,7,6,4,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => [6,5,4,3,7,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,3,6,2,1,7] => [5,6,4,7,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,3,6,5,2,1,7] => [5,4,7,6,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => [4,6,7,5,3,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [3,7,6,5,4,2,1] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [2,7,6,5,4,1,3] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [2,6,7,5,1,3,4] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,6,5,8,4,3,2] => [2,8,7,6,1,3,4,5] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [2,6,5,1,3,4,7] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [2,9,8,1,3,4,5,6,7] => ? ∊ {1,3,3,3,4,4,5,5,5,5,6,6,7,7,8,9,11}
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000674
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 82%
St000674: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 82%
Values
[1]
=> [1,0]
=> ? = 2 - 1
[2]
=> [1,0,1,0]
=> 2 = 3 - 1
[1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 8 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 9 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,2,3,4,5,6,7,8,10} - 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,9,11} - 1
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.
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000990The first ascent of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000247The number of singleton blocks of a set partition. St001571The Cartan determinant of the integer partition. St000022The number of fixed points of a permutation. St000974The length of the trunk of an ordered tree. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000993The multiplicity of the largest part of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St000873The aix statistic of a permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001933The largest multiplicity of a part in an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000215The number of adjacencies of a permutation, zero appended. St000895The number of ones on the main diagonal of an alternating sign matrix. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000461The rix statistic of a permutation. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St000315The number of isolated vertices of a graph. St000338The number of pixed points of a permutation. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000237The number of small exceedances. St000937The number of positive values of the symmetric group character corresponding to the partition. St000248The number of anti-singletons of a set partition. St001691The number of kings in a graph. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St000214The number of adjacencies of a permutation. St000456The monochromatic index of a connected graph. St000894The trace of an alternating sign matrix. St000335The difference of lower and upper interactions. St001060The distinguishing index of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000239The number of small weak excedances. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001903The number of fixed points of a parking function. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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