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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000160
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Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> 2
[4,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> 2
[2,1,1,1]
=> [1,1,1]
=> 3
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 1
[4,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> 3
[2,2,1,1]
=> [2,1,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> 4
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 1
[5,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> 3
[3,3,1]
=> [3,1]
=> 1
[3,2,2]
=> [2,2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> 4
[2,2,1,1,1]
=> [2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 1
[6,1,1]
=> [1,1]
=> 2
[5,3]
=> [3]
=> 1
[5,2,1]
=> [2,1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> 3
[4,3,1]
=> [3,1]
=> 1
[4,2,2]
=> [2,2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> 4
[3,3,1,1]
=> [3,1,1]
=> 2
[3,2,2,1]
=> [2,2,1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[2,2,2,1,1]
=> [2,2,1,1]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 4
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 6
[8,1]
=> [1]
=> 1
[7,2]
=> [2]
=> 1
[7,1,1]
=> [1,1]
=> 2
[6,3]
=> [3]
=> 1
[6,2,1]
=> [2,1]
=> 1
[5,3,1]
=> [3,1]
=> 1
[5,2,2]
=> [2,2]
=> 2
[5,2,1,1]
=> [2,1,1]
=> 2
[4,4,1]
=> [4,1]
=> 1
[4,3,2]
=> [3,2]
=> 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1] => 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,3] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1] => 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,3] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,1,2] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [4,1,1] => 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [1,1,5] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [2,1,3] => 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,1,2] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [4,1,1] => 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [3,2,1] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => [5,1,1] => 5
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,6] => 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [1,1,5] => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [2,1,4] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [1,1,1,3] => 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => [3,1,2] => 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1,2] => 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1] => 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => [4,1,1] => 4
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,1,1] => [3,1,1,1] => 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,5,1] => [5,1,1] => 5
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => [2,3,1] => 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,2] => [4,2,1] => 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,6,1] => [6,1,1] => 6
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1] => [1,1,7] => 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,6] => 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => [2,1,5] => 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,1] => [1,1,5] => 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,1,1,1] => [1,1,1,4] => 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,1,1,1] => [1,1,1,3] => 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,1] => [2,1,3] => 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,2,1,1] => [2,1,1,2] => 2
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [3,1,2] => [1,2,3] => 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000993
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> 1
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> 1
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [4]
=> 1
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [2]
=> 1
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [3,3]
=> 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [5]
=> 1
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [3]
=> 1
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [4,4]
=> 2
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [3,2]
=> 1
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 3
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [2]
=> 1
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> 2
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 4
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> 5
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [6]
=> 1
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [4]
=> 1
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [5,5]
=> 2
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [2]
=> 1
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [4,3]
=> 1
[5,1,1,1]
=> [[5,1,1,1],[]]
=> [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> 3
[4,3,1]
=> [[4,3,1],[]]
=> [[4,4,4],[3,1]]
=> [3,1]
=> 1
[4,2,2]
=> [[4,2,2],[]]
=> [[4,4,4],[2,2]]
=> [2,2]
=> 2
[4,2,1,1]
=> [[4,2,1,1],[]]
=> [[4,4,4,4],[3,3,2]]
=> [3,3,2]
=> 2
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> 4
[3,3,1,1]
=> [[3,3,1,1],[]]
=> [[3,3,3,3],[2,2]]
=> [2,2]
=> 2
[3,2,2,1]
=> [[3,2,2,1],[]]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> 1
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,1]]
=> [2,2,2,1]
=> 3
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [2,2,2,2,2]
=> 5
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> [[2,2,2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [1,1,1,1,1,1]
=> 6
[8,1]
=> [[8,1],[]]
=> [[8,8],[7]]
=> [7]
=> 1
[7,2]
=> [[7,2],[]]
=> [[7,7],[5]]
=> [5]
=> 1
[7,1,1]
=> [[7,1,1],[]]
=> [[7,7,7],[6,6]]
=> [6,6]
=> 2
[6,3]
=> [[6,3],[]]
=> [[6,6],[3]]
=> [3]
=> 1
[6,2,1]
=> [[6,2,1],[]]
=> [[6,6,6],[5,4]]
=> [5,4]
=> 1
[5,3,1]
=> [[5,3,1],[]]
=> [[5,5,5],[4,2]]
=> [4,2]
=> 1
[5,2,2]
=> [[5,2,2],[]]
=> [[5,5,5],[3,3]]
=> [3,3]
=> 2
[5,2,1,1]
=> [[5,2,1,1],[]]
=> [[5,5,5,5],[4,4,3]]
=> [4,4,3]
=> 2
[4,4,1]
=> [[4,4,1],[]]
=> [[4,4,4],[3]]
=> [3]
=> 1
[4,3,2]
=> [[4,3,2],[]]
=> [[4,4,4],[2,1]]
=> [2,1]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Mp00105: Binary words —complement⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[3,1]
=> 10010 => 01101 => 10101 => 1
[2,1,1]
=> 10110 => 01001 => 01001 => 2
[4,1]
=> 100010 => 011101 => 101101 => 1
[3,1,1]
=> 100110 => 011001 => 010101 => 2
[2,1,1,1]
=> 101110 => 010001 => 001001 => 3
[5,1]
=> 1000010 => 0111101 => 1011101 => 1
[4,2]
=> 100100 => 011011 => 101011 => 1
[4,1,1]
=> 1000110 => 0111001 => 0101101 => 2
[3,2,1]
=> 101010 => 010101 => 110001 => 1
[3,1,1,1]
=> 1001110 => 0110001 => 0010101 => 3
[2,2,1,1]
=> 110110 => 001001 => 010001 => 2
[2,1,1,1,1]
=> 1011110 => 0100001 => 0001001 => 4
[6,1]
=> 10000010 => 01111101 => 10111101 => 1
[5,2]
=> 1000100 => 0111011 => 1011011 => 1
[5,1,1]
=> 10000110 => 01111001 => 01011101 => 2
[4,2,1]
=> 1001010 => 0110101 => 1101001 => 1
[4,1,1,1]
=> 10001110 => 01110001 => 00101101 => 3
[3,3,1]
=> 110010 => 001101 => 100101 => 1
[3,2,2]
=> 101100 => 010011 => 010011 => 2
[3,2,1,1]
=> 1010110 => 0101001 => 0110001 => 2
[3,1,1,1,1]
=> 10011110 => 01100001 => 00010101 => 4
[2,2,1,1,1]
=> 1101110 => 0010001 => 0010001 => 3
[2,1,1,1,1,1]
=> 10111110 => 01000001 => 00001001 => 5
[7,1]
=> 100000010 => 011111101 => 101111101 => 1
[6,2]
=> 10000100 => 01111011 => 10111011 => 1
[6,1,1]
=> 100000110 => 011111001 => 010111101 => 2
[5,3]
=> 1001000 => 0110111 => 1010111 => 1
[5,2,1]
=> 10001010 => 01110101 => 11011001 => 1
[5,1,1,1]
=> 100001110 => 011110001 => 001011101 => 3
[4,3,1]
=> 1010010 => 0101101 => 1100101 => 1
[4,2,2]
=> 1001100 => 0110011 => 0101011 => 2
[4,2,1,1]
=> 10010110 => 01101001 => 01101001 => 2
[4,1,1,1,1]
=> 100011110 => 011100001 => 000101101 => 4
[3,3,1,1]
=> 1100110 => 0011001 => 0100101 => 2
[3,2,2,1]
=> 1011010 => 0100101 => 1010001 => 1
[3,2,1,1,1]
=> 10101110 => 01010001 => 00110001 => 3
[3,1,1,1,1,1]
=> 100111110 => 011000001 => 000010101 => 5
[2,2,2,1,1]
=> 1110110 => 0001001 => 0100001 => 2
[2,2,1,1,1,1]
=> 11011110 => 00100001 => 00010001 => 4
[2,1,1,1,1,1,1]
=> 101111110 => 010000001 => 000001001 => 6
[8,1]
=> 1000000010 => 0111111101 => 1011111101 => 1
[7,2]
=> 100000100 => 011111011 => 101111011 => 1
[7,1,1]
=> 1000000110 => 0111111001 => 0101111101 => 2
[6,3]
=> 10001000 => 01110111 => 10110111 => 1
[6,2,1]
=> 100001010 => 011110101 => 110111001 => 1
[5,3,1]
=> 10010010 => 01101101 => 11010101 => 1
[5,2,2]
=> 10001100 => 01110011 => 01011011 => 2
[5,2,1,1]
=> 100010110 => 011101001 => 011011001 => 2
[4,4,1]
=> 1100010 => 0011101 => 1001101 => 1
[4,3,2]
=> 1010100 => 0101011 => 1100011 => 1
[9,1]
=> 10000000010 => 01111111101 => 10111111101 => ? = 1
[8,2]
=> 1000000100 => 0111111011 => 1011111011 => ? = 1
[7,2,1]
=> 1000001010 => 0111110101 => 1101111001 => ? = 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01000000001 => 00000001001 => ? = 8
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: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Mp00278: Binary words —rowmotion⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Values
[3,1]
=> 10010 => 01100 => 00110 => 0 = 1 - 1
[2,1,1]
=> 10110 => 11001 => 10011 => 1 = 2 - 1
[4,1]
=> 100010 => 001100 => 001100 => 0 = 1 - 1
[3,1,1]
=> 100110 => 011001 => 100110 => 1 = 2 - 1
[2,1,1,1]
=> 101110 => 110011 => 110011 => 2 = 3 - 1
[5,1]
=> 1000010 => 0001100 => 0011000 => 0 = 1 - 1
[4,2]
=> 100100 => 011000 => 000110 => 0 = 1 - 1
[4,1,1]
=> 1000110 => 0011001 => 1001100 => 1 = 2 - 1
[3,2,1]
=> 101010 => 110100 => 001011 => 0 = 1 - 1
[3,1,1,1]
=> 1001110 => 0110011 => 1100110 => 2 = 3 - 1
[2,2,1,1]
=> 110110 => 111001 => 100111 => 1 = 2 - 1
[2,1,1,1,1]
=> 1011110 => 1100111 => 1110011 => 3 = 4 - 1
[6,1]
=> 10000010 => 00001100 => 00110000 => 0 = 1 - 1
[5,2]
=> 1000100 => 0011000 => 0001100 => 0 = 1 - 1
[5,1,1]
=> 10000110 => 00011001 => 10011000 => 1 = 2 - 1
[4,2,1]
=> 1001010 => 0110100 => 0010110 => 0 = 1 - 1
[4,1,1,1]
=> 10001110 => 00110011 => 11001100 => 2 = 3 - 1
[3,3,1]
=> 110010 => 011100 => 001110 => 0 = 1 - 1
[3,2,2]
=> 101100 => 110001 => 100011 => 1 = 2 - 1
[3,2,1,1]
=> 1010110 => 1101001 => 1001011 => 1 = 2 - 1
[3,1,1,1,1]
=> 10011110 => 01100111 => 11100110 => 3 = 4 - 1
[2,2,1,1,1]
=> 1101110 => 1110011 => 1100111 => 2 = 3 - 1
[2,1,1,1,1,1]
=> 10111110 => 11001111 => 11110011 => 4 = 5 - 1
[7,1]
=> 100000010 => 000001100 => 001100000 => 0 = 1 - 1
[6,2]
=> 10000100 => 00011000 => 00011000 => 0 = 1 - 1
[6,1,1]
=> 100000110 => 000011001 => 100110000 => 1 = 2 - 1
[5,3]
=> 1001000 => 0110000 => 0000110 => 0 = 1 - 1
[5,2,1]
=> 10001010 => 00110100 => 00101100 => 0 = 1 - 1
[5,1,1,1]
=> 100001110 => 000110011 => 110011000 => 2 = 3 - 1
[4,3,1]
=> 1010010 => 1100100 => 0010011 => 0 = 1 - 1
[4,2,2]
=> 1001100 => 0110001 => 1000110 => 1 = 2 - 1
[4,2,1,1]
=> 10010110 => 01101001 => 10010110 => 1 = 2 - 1
[4,1,1,1,1]
=> 100011110 => 001100111 => 111001100 => 3 = 4 - 1
[3,3,1,1]
=> 1100110 => 0111001 => 1001110 => 1 = 2 - 1
[3,2,2,1]
=> 1011010 => 1101100 => 0011011 => 0 = 1 - 1
[3,2,1,1,1]
=> 10101110 => 11010011 => 11001011 => 2 = 3 - 1
[3,1,1,1,1,1]
=> 100111110 => 011001111 => 111100110 => 4 = 5 - 1
[2,2,2,1,1]
=> 1110110 => 1111001 => 1001111 => 1 = 2 - 1
[2,2,1,1,1,1]
=> 11011110 => 11100111 => 11100111 => 3 = 4 - 1
[2,1,1,1,1,1,1]
=> 101111110 => 110011111 => 111110011 => 5 = 6 - 1
[8,1]
=> 1000000010 => 0000001100 => 0011000000 => ? = 1 - 1
[7,2]
=> 100000100 => 000011000 => 000110000 => 0 = 1 - 1
[7,1,1]
=> 1000000110 => 0000011001 => 1001100000 => ? = 2 - 1
[6,3]
=> 10001000 => 00110000 => 00001100 => 0 = 1 - 1
[6,2,1]
=> 100001010 => 000110100 => 001011000 => 0 = 1 - 1
[5,3,1]
=> 10010010 => 01100100 => 00100110 => 0 = 1 - 1
[5,2,2]
=> 10001100 => 00110001 => 10001100 => 1 = 2 - 1
[5,2,1,1]
=> 100010110 => 001101001 => 100101100 => 1 = 2 - 1
[4,4,1]
=> 1100010 => 0011100 => 0011100 => 0 = 1 - 1
[4,3,2]
=> 1010100 => 1101000 => 0001011 => 0 = 1 - 1
[4,3,1,1]
=> 10100110 => 11001001 => 10010011 => 1 = 2 - 1
[4,2,2,1]
=> 10011010 => 01101100 => 00110110 => 0 = 1 - 1
[3,1,1,1,1,1,1]
=> 1001111110 => 0110011111 => 1111100110 => ? = 6 - 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => 1100111111 => 1111110011 => ? = 7 - 1
[9,1]
=> 10000000010 => 00000001100 => 00110000000 => ? = 1 - 1
[3,2,1,1,1,1,1]
=> 1010111110 => 1101001111 => 1111001011 => ? = 5 - 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => 1110011111 => 1111100111 => ? = 6 - 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 11001111111 => 11111110011 => ? = 8 - 1
[5,4,2,1,1]
=> 1010010110 => 1100101001 => 1001010011 => ? = 2 - 1
[5,3,2,2,1]
=> 1001011010 => 0110101100 => 0011010110 => ? = 1 - 1
[5,4,3,1,1]
=> 1010100110 => 1101001001 => 1001001011 => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 3
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 4
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [1,1] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [1,1,1] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [2,1] => 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [1,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,1,2] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 4
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [2,3] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,5] => 5
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [1,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [1,1] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,2] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => [1,1] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,1,1] => 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => [1,3] => 3
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => [1,1,1] => 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => [1,2] => 2
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => [1,1,2] => 2
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => [1,4] => 4
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => [2,2] => 2
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => [1,2,1] => 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => [1,1,3] => 3
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => [1,5] => 5
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,1,1] => [3,2] => 2
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [2,2,1,1,1,1] => [2,4] => 4
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => [1,6] => 6
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [8,1] => [1,1] => 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [7,2] => [1,1] => 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [7,1,1] => [1,2] => 2
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [6,3] => [1,1] => 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [6,2,1] => [1,1,1] => 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [5,3,1] => [1,1,1] => 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [5,2,2] => [1,2] => 2
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [5,2,1,1] => [1,1,2] => 2
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [4,4,1] => [2,1] => 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => [1,1,1] => 1
[2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [2,2,1,1,1,1,1,1] => [2,6] => ? = 6
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? => ? => ? = 1
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [5,4,2] => ? => ? = 1
[5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [5,4,1,1] => ? => ? = 2
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [5,3,3] => ? => ? = 2
[5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [5,3,2,1] => ? => ? = 1
[5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [5,2,2,2] => ? => ? = 3
[4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [4,4,2,1] => ? => ? = 1
[4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [4,4,1,1,1] => ? => ? = 3
[4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [4,3,3,1] => ? => ? = 1
[4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? => ? = 2
[4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [4,3,2,1,1] => ? => ? = 2
[4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [4,2,2,2,1] => ? => ? = 1
[3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? => ? = 2
[3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? => ? = 1
[3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? => ? = 4
[5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? => ? = 1
[4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [4,4,2,1,1] => ? => ? = 2
[4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> [4,3,3,1,1] => ? => ? = 2
[3,3,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? => ? = 3
[5,5,3]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13]]
=> ? => ? => ? = 1
[5,4,4]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12,13]]
=> ? => ? => ? = 2
[5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> [5,4,3,1] => ? => ? = 1
[5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> [5,4,2,2] => ? => ? = 2
[5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [5,4,2,1,1] => ? => ? = 2
[5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [5,3,3,2] => ? => ? = 1
[5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [5,3,3,1,1] => ? => ? = 2
[5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [5,3,2,2,1] => ? => ? = 1
[4,4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13]]
=> ? => ? => ? = 1
[4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [4,4,3,2] => ? => ? = 1
[4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [4,4,3,1,1] => ? => ? = 2
[4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [4,4,2,2,1] => ? => ? = 1
[4,3,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13]]
=> ? => ? => ? = 3
[4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [4,3,3,2,1] => ? => ? = 1
[3,3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13]]
=> ? => ? => ? = 1
[3,3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13]]
=> ? => ? => ? = 2
[5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [5,4,3,2] => ? => ? = 1
[5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [5,4,3,1,1] => ? => ? = 2
[5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [5,4,2,2,1] => ? => ? = 1
[5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [5,3,3,2,1] => ? => ? = 1
[4,4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14]]
=> ? => ? => ? = 1
[4,4,3,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13,14]]
=> ? => ? => ? = 2
[4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [4,4,3,2,1] => ? => ? = 1
[5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [5,4,3,2,1] => ? => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000312
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 62%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 62%
Values
[3,1]
=> 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,1]
=> 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,1]
=> 100010 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,1,1]
=> 100110 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,1,1,1]
=> 101110 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[5,1]
=> 1000010 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,2]
=> 100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[4,1,1]
=> 1000110 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,1]
=> 101010 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,1,1,1]
=> 1001110 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[2,2,1,1]
=> 110110 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,1,1,1,1]
=> 1011110 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[6,1]
=> 10000010 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[5,2]
=> 1000100 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[5,1,1]
=> 10000110 => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 1 = 2 - 1
[4,2,1]
=> 1001010 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,1,1,1]
=> 10001110 => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 2 = 3 - 1
[3,3,1]
=> 110010 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,2,2]
=> 101100 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,1,1,1,1]
=> 10011110 => [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 4 - 1
[2,2,1,1,1]
=> 1101110 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[2,1,1,1,1,1]
=> 10111110 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 4 = 5 - 1
[7,1]
=> 100000010 => [1,6,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[6,2]
=> 10000100 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[6,1,1]
=> 100000110 => [1,5,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[5,3]
=> 1001000 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[5,2,1]
=> 10001010 => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[5,1,1,1]
=> 100001110 => [1,4,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[4,3,1]
=> 1010010 => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,2,2]
=> 1001100 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[4,2,1,1]
=> 10010110 => [1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 1 = 2 - 1
[4,1,1,1,1]
=> 100011110 => [1,3,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[3,3,1,1]
=> 1100110 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[3,2,2,1]
=> 1011010 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[3,2,1,1,1]
=> 10101110 => [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 2 = 3 - 1
[3,1,1,1,1,1]
=> 100111110 => [1,2,5,1] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[2,2,2,1,1]
=> 1110110 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 2 - 1
[2,2,1,1,1,1]
=> 11011110 => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 3 = 4 - 1
[2,1,1,1,1,1,1]
=> 101111110 => [1,1,6,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 6 - 1
[8,1]
=> 1000000010 => [1,7,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 - 1
[7,2]
=> 100000100 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[7,1,1]
=> 1000000110 => [1,6,2,1] => ([(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[6,3]
=> 10001000 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[6,2,1]
=> 100001010 => [1,4,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[5,3,1]
=> 10010010 => [1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[5,2,2]
=> 10001100 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 1 = 2 - 1
[5,2,1,1]
=> 100010110 => [1,3,1,1,2,1] => ([(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[4,4,1]
=> 1100010 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,3,2]
=> 1010100 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,3,1,1]
=> 10100110 => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 1 = 2 - 1
[4,2,2,1]
=> 10011010 => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[4,2,1,1,1]
=> 100101110 => [1,2,1,1,3,1] => ([(0,8),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[3,3,2,1]
=> 1101010 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[3,3,1,1,1]
=> 11001110 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 2 = 3 - 1
[3,2,2,2]
=> 1011100 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[3,2,2,1,1]
=> 10110110 => [1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 1 = 2 - 1
[3,2,1,1,1,1]
=> 101011110 => [1,1,1,1,4,1] => ([(0,8),(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[3,1,1,1,1,1,1]
=> 1001111110 => [1,2,6,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6 - 1
[2,2,2,1,1,1]
=> 11101110 => [3,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 2 = 3 - 1
[2,2,1,1,1,1,1]
=> 110111110 => [2,1,5,1] => ([(0,8),(1,8),(2,8),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => [1,1,7,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 7 - 1
[9,1]
=> 10000000010 => [1,8,1,1] => ([(0,9),(0,10),(1,9),(1,10),(2,9),(2,10),(3,9),(3,10),(4,9),(4,10),(5,9),(5,10),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 1 - 1
[8,2]
=> 1000000100 => [1,6,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 - 1
[7,3]
=> 100001000 => [1,4,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[7,2,1]
=> 1000001010 => [1,5,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1 - 1
[6,4]
=> 10010000 => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[6,3,1]
=> 100010010 => [1,3,1,2,1,1] => ([(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[6,2,2]
=> 100001100 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[5,4,1]
=> 10100010 => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[5,3,2]
=> 10010100 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[5,3,1,1]
=> 100100110 => [1,2,1,2,2,1] => ([(0,8),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[5,2,2,1]
=> 100011010 => [1,3,2,1,1,1] => ([(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[4,4,2]
=> 1100100 => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,3,1,1,1]
=> 101001110 => [1,1,1,2,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[4,2,2,1,1]
=> 100110110 => [1,2,2,1,2,1] => ([(0,8),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[3,3,1,1,1,1]
=> 110011110 => [2,2,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[3,2,2,1,1,1]
=> 101101110 => [1,1,2,1,3,1] => ([(0,8),(1,8),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[3,2,1,1,1,1,1]
=> 1010111110 => [1,1,1,1,5,1] => ([(0,9),(1,9),(2,9),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[2,2,2,1,1,1,1]
=> 111011110 => [3,1,4,1] => ([(0,8),(1,8),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => [2,1,6,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6 - 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => [1,1,8,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 8 - 1
[5,4,1,1]
=> 101000110 => [1,1,1,3,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[5,3,2,1]
=> 100101010 => [1,2,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[5,2,2,2]
=> 100011100 => [1,3,3,2] => ([(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[4,4,1,1,1]
=> 110001110 => [2,3,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[4,3,2,1,1]
=> 101010110 => [1,1,1,1,1,1,2,1] => ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[4,2,2,2,1]
=> 100111010 => [1,2,3,1,1,1] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[6,4,2]
=> 100100100 => [1,2,1,2,1,2] => ([(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[5,4,2,1]
=> 101001010 => [1,1,1,2,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[5,3,3,1]
=> 100110010 => [1,2,2,2,1,1] => ([(0,7),(0,8),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[5,3,2,2]
=> 100101100 => [1,2,1,1,2,2] => ([(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[4,4,2,1,1]
=> 110010110 => [2,2,1,1,2,1] => ([(0,8),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[4,3,3,1,1]
=> 101100110 => [1,1,2,2,2,1] => ([(0,8),(1,7),(1,8),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[4,3,2,2,1]
=> 101011010 => [1,1,1,1,2,1,1,1] => ([(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ([(0,8),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[5,4,3,1]
=> 101010010 => [1,1,1,1,1,2,1,1] => ([(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[5,4,2,2]
=> 101001100 => [1,1,1,2,2,2] => ([(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[5,4,2,1,1]
=> 1010010110 => [1,1,1,2,1,1,2,1] => ([(0,9),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[5,3,3,2]
=> 100110100 => [1,2,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 - 1
Description
The number of leaves in a graph.
That is, the number of vertices of a graph that have degree 1.
Matching statistic: St000907
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 62%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 62%
Values
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 3
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 4
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5)],6)
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ([(0,6),(6,1),(6,2),(6,3),(6,4),(6,5)],7)
=> 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ([(0,5),(5,6),(6,1),(6,2),(6,3),(6,4)],7)
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> 5
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ([(0,7),(7,1),(7,2),(7,3),(7,4),(7,5),(7,6)],8)
=> ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(2,3),(2,4),(2,5)],6)
=> 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ([(0,5),(0,6),(6,1),(6,2),(6,3),(6,4)],7)
=> 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(6,7),(7,1),(7,2),(7,3),(7,4),(7,5)],8)
=> ? = 3
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ([(0,5),(5,4),(5,6),(6,1),(6,2),(6,3)],7)
=> 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(5,7),(6,5),(7,1),(7,2),(7,3),(7,4)],8)
=> ? = 4
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,5),(4,3),(5,1),(5,2),(5,4)],6)
=> 2
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(4,6),(5,2),(5,3),(6,1),(6,5)],7)
=> 3
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(4,5),(5,7),(6,4),(7,1),(7,2),(7,3)],8)
=> ? = 5
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 6
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ? = 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ([(0,8),(8,1),(8,2),(8,3),(8,4),(8,5),(8,6),(8,7)],9)
=> ? = 2
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ([(2,3),(2,4),(2,5),(2,6)],7)
=> 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ([(0,6),(0,7),(7,1),(7,2),(7,3),(7,4),(7,5)],8)
=> ? = 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(0,6),(6,1),(6,2),(6,3)],7)
=> 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 2
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ([(0,6),(6,5),(6,7),(7,1),(7,2),(7,3),(7,4)],8)
=> ? = 2
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ([(0,6),(5,3),(5,4),(6,1),(6,2),(6,5)],7)
=> 2
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(2,5),(2,6),(3,1),(3,4),(3,5),(3,6)],7)
=> 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(5,6),(6,4),(6,7),(7,1),(7,2),(7,3)],8)
=> ? = 3
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ([(0,5),(4,3),(5,6),(6,1),(6,2),(6,4)],7)
=> 3
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> 3
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(2,6),(3,1),(3,5),(3,6),(4,2),(4,3)],7)
=> ? = 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(4,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 4
[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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(4,6),(5,4),(6,8),(7,5),(8,1),(8,2),(8,3)],9)
=> ? = 6
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 3
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 5
[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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(8,2)],9)
=> ? = 7
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9)],10)
=> ? = 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8)],9)
=> ? = 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ([(2,3),(2,4),(2,5),(2,6),(2,7)],8)
=> ? = 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> ([(0,7),(0,8),(8,1),(8,2),(8,3),(8,4),(8,5),(8,6)],9)
=> ? = 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ([(3,4),(3,5),(3,6)],7)
=> 1
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ([(0,5),(0,6),(0,7),(7,1),(7,2),(7,3),(7,4)],8)
=> ? = 1
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ? = 2
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
=> 1
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 1
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ([(0,7),(6,3),(6,4),(6,5),(7,1),(7,2),(7,6)],8)
=> ? = 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(2,5),(2,6),(2,7),(3,1),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ([(0,6),(5,4),(6,1),(6,2),(6,3),(6,5)],7)
=> 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 2
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(3,6),(4,1),(4,2),(4,5),(4,6)],7)
=> 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ([(0,5),(5,7),(6,3),(6,4),(7,1),(7,2),(7,6)],8)
=> ? = 3
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5)],7)
=> ? = 3
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,2),(4,3)],8)
=> ? = 2
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> 1
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
=> ? = 2
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(4,7),(5,3),(6,4),(7,1),(7,2),(7,5)],8)
=> ? = 4
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,1),(3,6)],7)
=> ? = 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,6),(2,7),(3,1),(3,6),(3,7),(4,5),(5,2),(5,3)],8)
=> ? = 3
[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]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(4,5),(5,7),(6,4),(7,1),(7,8),(8,2),(8,3)],9)
=> ? = 5
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
=> ? = 2
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4)],8)
=> ? = 4
[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]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(3,5),(4,3),(5,8),(6,1),(7,4),(8,2),(8,6)],9)
=> ? = 6
[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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,8),(3,5),(4,3),(5,7),(6,4),(7,9),(8,6),(9,1),(9,2)],10)
=> ? = 8
[5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 1
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> ([(0,7),(6,4),(6,5),(7,1),(7,2),(7,3),(7,6)],8)
=> ? = 2
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 2
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(3,6),(3,7),(4,1),(4,2),(4,5),(4,6),(4,7)],8)
=> ? = 1
[5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6)],8)
=> ? = 3
[4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(5,4),(6,7),(7,1),(7,2),(7,3),(7,5)],8)
=> ? = 3
[4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4)],7)
=> ? = 2
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ([(0,5),(3,6),(3,7),(4,1),(4,2),(4,6),(4,7),(5,3),(5,4)],8)
=> ? = 2
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> ([(0,1),(0,3),(1,2),(1,7),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
=> ? = 2
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000260
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 29%●distinct values known / distinct values provided: 12%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 29%●distinct values known / distinct values provided: 12%
Values
[3,1]
=> 11 => [2] => ([],2)
=> ? = 1
[2,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? = 2
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[3,1,1]
=> 111 => [3] => ([],3)
=> ? = 2
[2,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? = 3
[5,1]
=> 11 => [2] => ([],2)
=> ? = 1
[4,2]
=> 00 => [2] => ([],2)
=> ? = 1
[4,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? = 2
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,1,1]
=> 1111 => [4] => ([],4)
=> ? = 3
[2,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? = 4
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[5,1,1]
=> 111 => [3] => ([],3)
=> ? = 2
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[4,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? = 3
[3,3,1]
=> 111 => [3] => ([],3)
=> ? = 1
[3,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? = 2
[3,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[3,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? = 4
[2,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[2,1,1,1,1,1]
=> 011111 => [1,5] => ([(4,5)],6)
=> ? = 5
[7,1]
=> 11 => [2] => ([],2)
=> ? = 1
[6,2]
=> 00 => [2] => ([],2)
=> ? = 1
[6,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? = 2
[5,3]
=> 11 => [2] => ([],2)
=> ? = 1
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,1,1,1]
=> 1111 => [4] => ([],4)
=> ? = 3
[4,3,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? = 1
[4,2,2]
=> 000 => [3] => ([],3)
=> ? = 2
[4,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[4,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? = 4
[3,3,1,1]
=> 1111 => [4] => ([],4)
=> ? = 2
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3
[3,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? = 5
[2,2,2,1,1]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,2,1,1,1,1]
=> 001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? = 4
[2,1,1,1,1,1,1]
=> 0111111 => [1,6] => ([(5,6)],7)
=> ? = 6
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[7,1,1]
=> 111 => [3] => ([],3)
=> ? = 2
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[5,3,1]
=> 111 => [3] => ([],3)
=> ? = 1
[5,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? = 2
[5,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? = 2
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,3,1,1,1]
=> 11111 => [5] => ([],5)
=> ? = 3
[3,2,2,2]
=> 1000 => [1,3] => ([(2,3)],4)
=> ? = 3
[3,2,2,1,1]
=> 10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,2,1,1,1,1]
=> 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 4
[3,1,1,1,1,1,1]
=> 1111111 => [7] => ([],7)
=> ? = 6
[2,2,2,1,1,1]
=> 000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3
[2,2,1,1,1,1,1]
=> 0011111 => [2,5] => ([(4,6),(5,6)],7)
=> ? = 5
[2,1,1,1,1,1,1,1]
=> 01111111 => [1,7] => ([(6,7)],8)
=> ? = 7
[9,1]
=> 11 => [2] => ([],2)
=> ? = 1
[8,2]
=> 00 => [2] => ([],2)
=> ? = 1
[7,3]
=> 11 => [2] => ([],2)
=> ? = 1
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,4,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[3,3,2,2,1]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,5,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[5,4,3]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,4,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,3,2]
=> 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,2,1]
=> 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,3,3,2,1]
=> 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,3,3,2]
=> 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[5,3,2,2,1]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,4,4,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,4,3,2]
=> 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,4,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[4,3,3,2,1]
=> 01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,3,2]
=> 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5,4,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,3,3,2,1]
=> 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,4,3,2,1]
=> 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,3,2,1]
=> 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000371
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 1 = 2 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 2 = 3 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0 = 1 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,4,2,3,5,6] => 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 1 = 2 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,4,5,2,3,6] => 0 = 1 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 2 = 3 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,5,4,6,2,3] => 1 = 2 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 3 = 4 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0 = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,4,2,3,5,6,7] => 0 = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 1 = 2 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,4,5,2,3,6,7] => 0 = 1 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 2 = 3 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,4,2,7,3,5,6] => 0 = 1 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,6,2,5,3,4,7] => ? = 2 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,5,4,6,2,3,7] => ? = 2 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 3 = 4 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,6,5,4,7,2,3] => ? = 3 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 4 = 5 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 0 = 1 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,4,2,3,5,6,7,8] => 0 = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 1 = 2 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,6,3,4,5,7,8] => ? = 1 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? = 1 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 2 = 3 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,4,2,7,3,5,6,8] => 0 = 1 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,6,2,5,3,4,7,8] => 1 = 2 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,5,4,6,2,3,7,8] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 3 = 4 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,5,4,2,8,3,6,7] => 1 = 2 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,3,7,5,6,2,4,8] => ? = 1 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,6,5,4,7,2,3,8] => ? = 3 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 4 = 5 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,3,7,8,5,6,2,4] => ? = 2 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,7,6,5,4,8,2,3] => 3 = 4 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 5 = 6 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 0 = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,4,2,3,5,6,7,8,9] => ? = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => 1 = 2 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,6,3,4,5,7,8,9] => ? = 1 - 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,4,5,2,3,6,7,8,9] => ? = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,4,2,7,3,5,6,8,9] => ? = 1 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,6,2,5,3,4,7,8,9] => ? = 2 - 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,5,4,6,2,3,7,8,9] => ? = 2 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,4,2,3,9,5,6,7,8] => ? = 1 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,2,4,7,8,3,5,6,9] => ? = 1 - 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [1,5,4,2,8,3,6,7,9] => ? = 2 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [1,3,7,5,6,2,4,8,9] => ? = 1 - 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [1,6,5,4,7,2,3,8,9] => ? = 3 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [1,3,2,7,8,9,4,5,6] => ? = 1 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [1,6,5,4,2,9,3,7,8] => ? = 3 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,8,2,7,3,6,4,5,9] => ? = 3 - 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,3,7,8,5,6,2,4,9] => ? = 2 - 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [1,7,6,5,4,8,2,3,9] => ? = 4 - 1
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => 5 = 6 - 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [1,3,8,7,9,5,6,2,4] => ? = 3 - 1
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,9,1,7] => [1,8,7,6,5,4,9,2,3] => ? = 5 - 1
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => 6 = 7 - 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,4,2,3,5,6,7,8,9,10] => ? = 1 - 1
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,2,6,3,4,5,7,8,9,10] => ? = 1 - 1
[7,2,1]
=> [[1,3,6,7,8,9,10],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9,10] => [1,4,5,2,3,6,7,8,9,10] => ? = 1 - 1
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,8,4,5,6,7,9,10] => ? = 1 - 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [1,4,2,7,3,5,6,8,9,10] => ? = 1 - 1
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [1,6,2,5,3,4,7,8,9,10] => ? = 2 - 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [1,4,2,3,9,5,6,7,8,10] => ? = 1 - 1
[5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9,10] => [1,2,4,7,8,3,5,6,9,10] => ? = 1 - 1
[5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9,10] => [1,5,4,2,8,3,6,7,9,10] => ? = 2 - 1
[5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9,10] => [1,3,7,5,6,2,4,8,9,10] => ? = 1 - 1
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6] => [1,2,4,7,3,10,5,6,8,9] => ? = 1 - 1
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6] => [1,5,4,2,3,10,6,7,8,9] => ? = 2 - 1
[4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10] => [1,2,9,3,4,8,5,6,7,10] => ? = 2 - 1
[4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10] => [1,3,2,7,8,9,4,5,6,10] => ? = 1 - 1
[4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10] => [1,6,5,4,2,9,3,7,8,10] => ? = 3 - 1
[4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10] => [1,8,2,7,3,6,4,5,9,10] => ? = 3 - 1
[4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10] => [1,3,7,8,5,6,2,4,9,10] => ? = 2 - 1
[3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4] => [1,3,2,10,6,4,9,5,7,8] => ? = 1 - 1
[3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7] => [1,2,4,9,5,8,10,3,6,7] => ? = 2 - 1
[3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7] => [1,3,7,2,8,9,10,4,5,6] => ? = 2 - 1
[3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [8,5,4,3,2,9,10,1,6,7] => [1,7,6,5,4,2,10,3,8,9] => ? = 4 - 1
[3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10] => [1,3,9,4,8,6,7,2,5,10] => ? = 1 - 1
[3,2,2,1,1,1]
=> [[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5,10] => [1,3,8,7,9,5,6,2,4,10] => ? = 3 - 1
[2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => 7 = 8 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000366The number of double descents of a permutation. St001826The maximal number of leaves on a vertex of a graph. St000441The number of successions of a permutation. St000546The number of global descents of a permutation. St000451The length of the longest pattern of the form k 1 2. St000925The number of topologically connected components of a set partition. St000028The number of stack-sorts needed to sort a permutation. St000731The number of double exceedences of a permutation. St001479The number of bridges of a graph. St000846The maximal number of elements covering an element of a poset. St000153The number of adjacent cycles of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000007The number of saliances of the permutation. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000365The number of double ascents of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000215The number of adjacencies of a permutation, zero appended. St000910The number of maximal chains of minimal length in a poset. St001399The distinguishing number of a poset. St000308The height of the tree associated to a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral.
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