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Your data matches 55 different statistics following compositions of up to 3 maps.
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Matching statistic: St000475
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(load all 5 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]
=> 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
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 0
[6,1,1]
=> 2
[5,3]
=> 0
[5,2,1]
=> 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)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Mp00105: Binary words —complement⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Values
[1]
=> 1 => 0 => 0 => 2 = 1 + 1
[2]
=> 0 => 1 => 1 => 1 = 0 + 1
[1,1]
=> 11 => 00 => 00 => 3 = 2 + 1
[3]
=> 1 => 0 => 0 => 2 = 1 + 1
[2,1]
=> 01 => 10 => 10 => 1 = 0 + 1
[1,1,1]
=> 111 => 000 => 000 => 4 = 3 + 1
[4]
=> 0 => 1 => 1 => 1 = 0 + 1
[3,1]
=> 11 => 00 => 00 => 3 = 2 + 1
[2,2]
=> 00 => 11 => 11 => 1 = 0 + 1
[2,1,1]
=> 011 => 100 => 010 => 2 = 1 + 1
[1,1,1,1]
=> 1111 => 0000 => 0000 => 5 = 4 + 1
[5]
=> 1 => 0 => 0 => 2 = 1 + 1
[4,1]
=> 01 => 10 => 10 => 1 = 0 + 1
[3,2]
=> 10 => 01 => 01 => 2 = 1 + 1
[3,1,1]
=> 111 => 000 => 000 => 4 = 3 + 1
[2,2,1]
=> 001 => 110 => 110 => 1 = 0 + 1
[2,1,1,1]
=> 0111 => 1000 => 0010 => 3 = 2 + 1
[1,1,1,1,1]
=> 11111 => 00000 => 00000 => 6 = 5 + 1
[6]
=> 0 => 1 => 1 => 1 = 0 + 1
[5,1]
=> 11 => 00 => 00 => 3 = 2 + 1
[4,2]
=> 00 => 11 => 11 => 1 = 0 + 1
[4,1,1]
=> 011 => 100 => 010 => 2 = 1 + 1
[3,3]
=> 11 => 00 => 00 => 3 = 2 + 1
[3,2,1]
=> 101 => 010 => 100 => 1 = 0 + 1
[3,1,1,1]
=> 1111 => 0000 => 0000 => 5 = 4 + 1
[2,2,2]
=> 000 => 111 => 111 => 1 = 0 + 1
[2,2,1,1]
=> 0011 => 1100 => 0110 => 2 = 1 + 1
[2,1,1,1,1]
=> 01111 => 10000 => 00010 => 4 = 3 + 1
[1,1,1,1,1,1]
=> 111111 => 000000 => 000000 => 7 = 6 + 1
[7]
=> 1 => 0 => 0 => 2 = 1 + 1
[6,1]
=> 01 => 10 => 10 => 1 = 0 + 1
[5,2]
=> 10 => 01 => 01 => 2 = 1 + 1
[5,1,1]
=> 111 => 000 => 000 => 4 = 3 + 1
[4,3]
=> 01 => 10 => 10 => 1 = 0 + 1
[4,2,1]
=> 001 => 110 => 110 => 1 = 0 + 1
[4,1,1,1]
=> 0111 => 1000 => 0010 => 3 = 2 + 1
[3,3,1]
=> 111 => 000 => 000 => 4 = 3 + 1
[3,2,2]
=> 100 => 011 => 011 => 2 = 1 + 1
[3,2,1,1]
=> 1011 => 0100 => 0100 => 2 = 1 + 1
[3,1,1,1,1]
=> 11111 => 00000 => 00000 => 6 = 5 + 1
[2,2,2,1]
=> 0001 => 1110 => 1110 => 1 = 0 + 1
[2,2,1,1,1]
=> 00111 => 11000 => 00110 => 3 = 2 + 1
[2,1,1,1,1,1]
=> 011111 => 100000 => 000010 => 5 = 4 + 1
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0000000 => 8 = 7 + 1
[8]
=> 0 => 1 => 1 => 1 = 0 + 1
[7,1]
=> 11 => 00 => 00 => 3 = 2 + 1
[6,2]
=> 00 => 11 => 11 => 1 = 0 + 1
[6,1,1]
=> 011 => 100 => 010 => 2 = 1 + 1
[5,3]
=> 11 => 00 => 00 => 3 = 2 + 1
[5,2,1]
=> 101 => 010 => 100 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => 00000000000 => ? = 11 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => 000000000000 => ? = 12 + 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,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000877
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(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: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Mp00105: Binary words —complement⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Values
[1]
=> 1 => 0 => 1
[2]
=> 0 => 1 => 0
[1,1]
=> 11 => 00 => 2
[3]
=> 1 => 0 => 1
[2,1]
=> 01 => 10 => 0
[1,1,1]
=> 111 => 000 => 3
[4]
=> 0 => 1 => 0
[3,1]
=> 11 => 00 => 2
[2,2]
=> 00 => 11 => 0
[2,1,1]
=> 011 => 100 => 1
[1,1,1,1]
=> 1111 => 0000 => 4
[5]
=> 1 => 0 => 1
[4,1]
=> 01 => 10 => 0
[3,2]
=> 10 => 01 => 1
[3,1,1]
=> 111 => 000 => 3
[2,2,1]
=> 001 => 110 => 0
[2,1,1,1]
=> 0111 => 1000 => 2
[1,1,1,1,1]
=> 11111 => 00000 => 5
[6]
=> 0 => 1 => 0
[5,1]
=> 11 => 00 => 2
[4,2]
=> 00 => 11 => 0
[4,1,1]
=> 011 => 100 => 1
[3,3]
=> 11 => 00 => 2
[3,2,1]
=> 101 => 010 => 1
[3,1,1,1]
=> 1111 => 0000 => 4
[2,2,2]
=> 000 => 111 => 0
[2,2,1,1]
=> 0011 => 1100 => 0
[2,1,1,1,1]
=> 01111 => 10000 => 3
[1,1,1,1,1,1]
=> 111111 => 000000 => 6
[7]
=> 1 => 0 => 1
[6,1]
=> 01 => 10 => 0
[5,2]
=> 10 => 01 => 1
[5,1,1]
=> 111 => 000 => 3
[4,3]
=> 01 => 10 => 0
[4,2,1]
=> 001 => 110 => 0
[4,1,1,1]
=> 0111 => 1000 => 2
[3,3,1]
=> 111 => 000 => 3
[3,2,2]
=> 100 => 011 => 1
[3,2,1,1]
=> 1011 => 0100 => 2
[3,1,1,1,1]
=> 11111 => 00000 => 5
[2,2,2,1]
=> 0001 => 1110 => 0
[2,2,1,1,1]
=> 00111 => 11000 => 1
[2,1,1,1,1,1]
=> 011111 => 100000 => 4
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 7
[8]
=> 0 => 1 => 0
[7,1]
=> 11 => 00 => 2
[6,2]
=> 00 => 11 => 0
[6,1,1]
=> 011 => 100 => 1
[5,3]
=> 11 => 00 => 2
[5,2,1]
=> 101 => 010 => 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => ? = 11
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1100000000 => ? ∊ {6,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => ? ∊ {6,12}
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 7 compositions to match this statistic)
(load all 7 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: 77% ●values known / values provided: 98%●distinct values known / distinct values provided: 77%
Mp00278: Binary words —rowmotion⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 98%●distinct values known / distinct values provided: 77%
Values
[1]
=> 1 => 1 => 1 => 1
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 2
[3]
=> 1 => 1 => 1 => 1
[2,1]
=> 01 => 10 => 01 => 0
[1,1,1]
=> 111 => 111 => 111 => 3
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 2
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 101 => 101 => 1
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 10 => 01 => 0
[3,2]
=> 10 => 01 => 10 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 010 => 010 => 0
[2,1,1,1]
=> 0111 => 1011 => 1101 => 2
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 2
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 101 => 101 => 1
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 110 => 011 => 0
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0101 => 1010 => 1
[2,1,1,1,1]
=> 01111 => 10111 => 11101 => 3
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
[7]
=> 1 => 1 => 1 => 1
[6,1]
=> 01 => 10 => 01 => 0
[5,2]
=> 10 => 01 => 10 => 1
[5,1,1]
=> 111 => 111 => 111 => 3
[4,3]
=> 01 => 10 => 01 => 0
[4,2,1]
=> 001 => 010 => 010 => 0
[4,1,1,1]
=> 0111 => 1011 => 1101 => 2
[3,3,1]
=> 111 => 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 100 => 1
[3,2,1,1]
=> 1011 => 1101 => 1011 => 1
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 0010 => 0100 => 0
[2,2,1,1,1]
=> 00111 => 01011 => 11010 => 2
[2,1,1,1,1,1]
=> 011111 => 101111 => 111101 => 4
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7
[8]
=> 0 => 0 => 0 => 0
[7,1]
=> 11 => 11 => 11 => 2
[6,2]
=> 00 => 00 => 00 => 0
[6,1,1]
=> 011 => 101 => 101 => 1
[5,3]
=> 11 => 11 => 11 => 2
[5,2,1]
=> 101 => 110 => 011 => 0
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => ? => ? => ? = 11
[3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {7,10,12}
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? => ? => ? ∊ {7,10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 111111111111 => ? ∊ {7,10,12}
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: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 85%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 85%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 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 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1]
=> [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 = 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 = 4 + 1
[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 + 1
[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 = 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
[1,1,1,1]
=> [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 = 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 = 5 + 1
[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 + 1
[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 = 1 + 1
[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 + 1
[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 = 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
[1,1,1,1,1]
=> [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 = 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 = 6 + 1
[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 + 1
[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 = 2 + 1
[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 + 1
[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 = 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 = 1 + 1
[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 + 1
[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 = 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 = 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
[1,1,1,1,1,1]
=> [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 = 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 = 7 + 1
[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 + 1
[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 = 3 + 1
[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 + 1
[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 = 1 + 1
[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 = 2 + 1
[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 + 1
[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 = 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 = 1 + 1
[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 = 1 + 1
[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 + 1
[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 = 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 = 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
[1,1,1,1,1,1,1]
=> [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 = 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 = 8 + 1
[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 + 1
[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 = 4 + 1
[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 + 1
[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 = 2 + 1
[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
[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]
=> ? = 2 + 1
[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]
=> ? ∊ {0,1,2} + 1
[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]
=> ? ∊ {0,1,2} + 1
[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]
=> ? ∊ {0,1,2} + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,1,1,1,2,3,4,5,8,11} + 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4,4,5,6,6,7,8,8,9,10,12} + 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: 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: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 77%
Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 77%
Values
[1]
=> 10 => 00 => [2] => 2 = 1 + 1
[2]
=> 100 => 000 => [3] => 3 = 2 + 1
[1,1]
=> 110 => 010 => [1,1,1] => 1 = 0 + 1
[3]
=> 1000 => 0000 => [4] => 4 = 3 + 1
[2,1]
=> 1010 => 0010 => [2,1,1] => 2 = 1 + 1
[1,1,1]
=> 1110 => 0110 => [1,2,1] => 1 = 0 + 1
[4]
=> 10000 => 00000 => [5] => 5 = 4 + 1
[3,1]
=> 10010 => 00010 => [3,1,1] => 3 = 2 + 1
[2,2]
=> 1100 => 0100 => [1,1,2] => 1 = 0 + 1
[2,1,1]
=> 10110 => 00110 => [2,2,1] => 2 = 1 + 1
[1,1,1,1]
=> 11110 => 01110 => [1,3,1] => 1 = 0 + 1
[5]
=> 100000 => 000000 => [6] => 6 = 5 + 1
[4,1]
=> 100010 => 000010 => [4,1,1] => 4 = 3 + 1
[3,2]
=> 10100 => 00100 => [2,1,2] => 2 = 1 + 1
[3,1,1]
=> 100110 => 000110 => [3,2,1] => 3 = 2 + 1
[2,2,1]
=> 11010 => 01010 => [1,1,1,1,1] => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 001110 => [2,3,1] => 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => 011110 => [1,4,1] => 1 = 0 + 1
[6]
=> 1000000 => 0000000 => [7] => 7 = 6 + 1
[5,1]
=> 1000010 => 0000010 => [5,1,1] => 5 = 4 + 1
[4,2]
=> 100100 => 000100 => [3,1,2] => 3 = 2 + 1
[4,1,1]
=> 1000110 => 0000110 => [4,2,1] => 4 = 3 + 1
[3,3]
=> 11000 => 01000 => [1,1,3] => 1 = 0 + 1
[3,2,1]
=> 101010 => 001010 => [2,1,1,1,1] => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0001110 => [3,3,1] => 3 = 2 + 1
[2,2,2]
=> 11100 => 01100 => [1,2,2] => 1 = 0 + 1
[2,2,1,1]
=> 110110 => 010110 => [1,1,1,2,1] => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0011110 => [2,4,1] => 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111110 => [1,5,1] => 1 = 0 + 1
[7]
=> 10000000 => 00000000 => [8] => 8 = 7 + 1
[6,1]
=> 10000010 => 00000010 => [6,1,1] => 6 = 5 + 1
[5,2]
=> 1000100 => 0000100 => [4,1,2] => 4 = 3 + 1
[5,1,1]
=> 10000110 => 00000110 => [5,2,1] => 5 = 4 + 1
[4,3]
=> 101000 => 001000 => [2,1,3] => 2 = 1 + 1
[4,2,1]
=> 1001010 => 0001010 => [3,1,1,1,1] => 3 = 2 + 1
[4,1,1,1]
=> 10001110 => 00001110 => [4,3,1] => 4 = 3 + 1
[3,3,1]
=> 110010 => 010010 => [1,1,2,1,1] => 1 = 0 + 1
[3,2,2]
=> 101100 => 001100 => [2,2,2] => 2 = 1 + 1
[3,2,1,1]
=> 1010110 => 0010110 => [2,1,1,2,1] => 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => 00011110 => [3,4,1] => 3 = 2 + 1
[2,2,2,1]
=> 111010 => 011010 => [1,2,1,1,1] => 1 = 0 + 1
[2,2,1,1,1]
=> 1101110 => 0101110 => [1,1,1,3,1] => 1 = 0 + 1
[2,1,1,1,1,1]
=> 10111110 => 00111110 => [2,5,1] => 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 01111110 => [1,6,1] => 1 = 0 + 1
[8]
=> 100000000 => 000000000 => [9] => 9 = 8 + 1
[7,1]
=> 100000010 => 000000010 => [7,1,1] => 7 = 6 + 1
[6,2]
=> 10000100 => 00000100 => [5,1,2] => 5 = 4 + 1
[6,1,1]
=> 100000110 => 000000110 => [6,2,1] => 6 = 5 + 1
[5,3]
=> 1001000 => 0001000 => [3,1,3] => 3 = 2 + 1
[5,2,1]
=> 10001010 => 00001010 => [4,1,1,1,1] => 4 = 3 + 1
[3,1,1,1,1,1,1]
=> 1001111110 => 0001111110 => [3,6,1] => ? ∊ {1,2} + 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0011111110 => [2,7,1] => ? ∊ {1,2} + 1
[10]
=> 10000000000 => 00000000000 => [11] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[9,1]
=> 10000000010 => 00000000010 => [9,1,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[8,2]
=> 1000000100 => 0000000100 => [7,1,2] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[8,1,1]
=> 10000000110 => 00000000110 => [8,2,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[7,1,1,1]
=> 10000001110 => 00000001110 => [7,3,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[6,2,1,1]
=> 1000010110 => 0000010110 => [5,1,1,2,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[6,1,1,1,1]
=> 10000011110 => 00000011110 => [6,4,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[5,2,1,1,1]
=> 1000101110 => 0000101110 => [4,1,1,3,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[5,1,1,1,1,1]
=> 10000111110 => 00000111110 => [5,5,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[4,2,1,1,1,1]
=> 1001011110 => 0001011110 => [3,1,1,4,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[4,1,1,1,1,1,1]
=> 10001111110 => 00001111110 => [4,6,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[3,2,1,1,1,1,1]
=> 1010111110 => 0010111110 => [2,1,1,5,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => 00011111110 => [3,7,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => 0101111110 => [1,1,1,6,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 00111111110 => [2,8,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 01111111110 => [1,9,1] => ? ∊ {0,0,1,1,2,2,3,3,4,4,5,6,6,7,8,10} + 1
[11]
=> 100000000000 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[10,1]
=> 100000000010 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[9,2]
=> 10000000100 => 00000000100 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[9,1,1]
=> 100000000110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[8,3]
=> 1000001000 => 0000001000 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[8,2,1]
=> 10000001010 => 00000001010 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[8,1,1,1]
=> 100000001110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[7,3,1]
=> 1000010010 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[7,2,2]
=> 1000001100 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[7,2,1,1]
=> 10000010110 => 00000010110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[6,3,1,1]
=> 1000100110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[6,2,2,1]
=> 1000011010 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[6,2,1,1,1]
=> 10000101110 => 00000101110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[6,1,1,1,1,1]
=> 100000111110 => 000000111110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[5,3,1,1,1]
=> 1001001110 => 0001001110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[5,2,2,1,1]
=> 1000110110 => 0000110110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[5,2,1,1,1,1]
=> 10001011110 => 00001011110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[4,2,1,1,1,1,1]
=> 10010111110 => 00010111110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => 00101111110 => [2,1,1,6,1] => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0110111110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01011111110 => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[12]
=> 1000000000000 => ? => ? => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
Description
The first part of an integer composition.
Matching statistic: St001176
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 69%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 69%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,2}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1,3}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,4}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,2,4}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,2,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,3,5}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,2,3,5}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,2,3,5}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,2,3,5}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,2,3,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,3,4,6}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,2,3,4,6}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,2,3,4,6}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,3,4,6}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,2,3,4,6}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,3,4,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,2,3,4,6}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,2,3,4,5,7}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,1,2,3,4,5,7}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,2,3,4,5,7}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,3,4,5,7}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,2,3,4,5,7}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,3,4,5,7}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,3,4,5,7}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,2,3,4,5,7}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2,3,3,4,5,6,7,9}
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 5
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001714
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 69%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 69%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,2}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1,3}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,4}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,2,4}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,2,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,3,5}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,2,3,5}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,2,3,5}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,2,3,5}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,2,3,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,3,4,6}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,2,3,4,6}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,2,3,4,6}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,3,4,6}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,2,3,4,6}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,3,4,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,2,3,4,6}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,2,3,4,5,7}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,1,2,3,4,5,7}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,2,3,4,5,7}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,3,4,5,7}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,2,3,4,5,7}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,3,4,5,7}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,2,3,4,5,7}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,2,3,4,5,7}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,1,2,2,3,4,5,6,8}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,2,3,3,4,5,6,7,9}
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 5
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic 0 are wide partitions.
Matching statistic: St000674
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 69%
St000674: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 69%
Values
[1]
=> [1,0]
=> ? = 1
[2]
=> [1,0,1,0]
=> 2
[1,1]
=> [1,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 3
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[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]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[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]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[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]
=> ? ∊ {0,0,1,2,3,4,5,6,7,9}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[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]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[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]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[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]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[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]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[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]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[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]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[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]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,8,10}
[11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
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: 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: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 85%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 85%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [2,1] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,3,2,1] => 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,4,2,1] => 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 1 = 0 + 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 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => 4 = 3 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 1 = 0 + 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
[1,1,1,1,1]
=> [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 = 0 + 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 = 6 + 1
[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 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => 3 = 2 + 1
[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 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [4,3,2,1,5] => 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[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 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => 1 = 0 + 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 = 0 + 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
[1,1,1,1,1,1]
=> [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 = 0 + 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 = 7 + 1
[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] => ? = 5 + 1
[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 = 3 + 1
[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 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => 3 = 2 + 1
[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 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => 1 = 0 + 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 = 1 + 1
[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 = 1 + 1
[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 + 1
[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 = 0 + 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 = 0 + 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
[1,1,1,1,1,1,1]
=> [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 = 0 + 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 = 8 + 1
[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] => ? ∊ {2,4,5,6} + 1
[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] => ? ∊ {2,4,5,6} + 1
[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] => ? ∊ {2,4,5,6} + 1
[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 = 2 + 1
[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 = 3 + 1
[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 + 1
[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 = 0 + 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 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 3 = 2 + 1
[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] => ? ∊ {2,4,5,6} + 1
[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] => ? ∊ {2,3,3,4,5,6,7} + 1
[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] => ? ∊ {2,3,3,4,5,6,7} + 1
[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] => ? ∊ {2,3,3,4,5,6,7} + 1
[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] => ? ∊ {2,3,3,4,5,6,7} + 1
[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] => ? ∊ {2,3,3,4,5,6,7} + 1
[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] => ? ∊ {2,3,3,4,5,6,7} + 1
[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] => ? ∊ {2,3,3,4,5,6,7} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[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] => ? ∊ {0,2,2,2,3,3,4,4,4,4,5,5,6,6,7,8,10} + 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,8,7,6,5,4,3,2,1,11] => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,7,9,6,5,4,3,2,1,10] => [9,8,10,7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,8,6,5,4,3,2,1,10] => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [7,6,5,8,4,3,2,1,9] => [8,7,6,9,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,8,5,4,3,2,1,9] => [7,8,9,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,7,6,4,3,2,1,9] => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [5,6,4,7,3,2,1,8] => [6,7,5,8,4,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,5,3,2,1,8] => [5,7,8,6,4,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,6,5,4,2,1,8] => ? => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [6,5,4,3,2,7,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,3,2,6,1,7] => [5,6,4,3,7,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [4,3,5,6,2,1,7] => [5,4,6,7,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,5,4,6,2,1,7] => [4,6,5,7,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [3,4,6,5,2,1,7] => [4,5,7,6,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,4,3,1,7] => [3,6,7,5,4,2,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [2,7,6,5,4,3,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => [2,6,7,5,4,1,3] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,7,6,5,4,8,3,2] => [2,8,7,6,5,1,3,4] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => [2,6,5,7,1,3,4] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => [2,5,7,6,1,3,4] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,6,7,5,8,4,3,2] => [2,7,8,6,1,3,4,5] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9} + 1
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].
The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000247The number of singleton blocks of a set partition. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000022The number of fixed points of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St000873The aix statistic of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000461The rix statistic of a permutation. St000215The number of adjacencies of a permutation, zero appended. 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. St000237The number of small exceedances. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001498The normalised height of a Nakayama algebra with magnitude 1. St000248The number of anti-singletons of a set partition. St001691The number of kings in a graph. St000214The number of adjacencies of a permutation. 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. St000894The trace of an alternating sign matrix. St000239The number of small weak excedances. 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. St000335The difference of lower and upper interactions. 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. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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.
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