Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000054
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [2,1] => 2
[2,1] => [1,2] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 2
[2,1,3] => [1,3,2] => [3,1,2] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [3,2,1] => [3,2,1] => 3
[3,1,2] => [1,2,3] => [3,2,1] => [3,2,1] => 3
[3,2,1] => [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [2,4,3,1] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [3,1,4,2] => 3
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [3,4,2,1] => 3
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [3,2,4,1] => 3
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [3,2,4,1] => 3
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [3,4,2,1] => 3
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [3,2,4,1] => 3
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [3,2,4,1] => 3
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [3,1,4,2] => 3
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [2,4,3,1] => 2
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [2,4,3,1] => 2
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [3,2,4,1] => 3
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [3,2,4,1] => 3
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [2,4,3,1] => 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [3,1,4,2] => 3
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [3,1,4,2] => 3
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [2,5,4,3,1] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [3,1,5,4,2] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [3,5,4,2,1] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [3,2,5,4,1] => 3
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [3,2,5,4,1] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [4,2,1,5,3] => 4
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [2,4,1,5,3] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [4,1,5,3,2] => 4
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [4,5,3,2,1] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [4,2,5,3,1] => 4
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [4,2,5,3,1] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [4,3,1,5,2] => 4
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [3,4,1,5,2] => 3
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [3,4,1,5,2] => 3
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [4,3,1,5,2] => 4
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [4,3,5,2,1] => 4
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Matching statistic: St000297
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 100%●distinct values known / distinct values provided: 82%
Values
[1] => [1] => [1] => => ? = 1 - 1
[1,2] => [1,2] => [2,1] => 1 => 1 = 2 - 1
[2,1] => [1,2] => [2,1] => 1 => 1 = 2 - 1
[1,2,3] => [1,2,3] => [3,2,1] => 11 => 2 = 3 - 1
[1,3,2] => [1,3,2] => [3,1,2] => 10 => 1 = 2 - 1
[2,1,3] => [1,3,2] => [3,1,2] => 10 => 1 = 2 - 1
[2,3,1] => [1,2,3] => [3,2,1] => 11 => 2 = 3 - 1
[3,1,2] => [1,2,3] => [3,2,1] => 11 => 2 = 3 - 1
[3,2,1] => [1,2,3] => [3,2,1] => 11 => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 111 => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 101 => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 110 => 2 = 3 - 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 110 => 2 = 3 - 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 110 => 2 = 3 - 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 110 => 2 = 3 - 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 110 => 2 = 3 - 1
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => 110 => 2 = 3 - 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 110 => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 111 => 3 = 4 - 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 110 => 2 = 3 - 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 101 => 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 101 => 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => 110 => 2 = 3 - 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 110 => 2 = 3 - 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 101 => 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 111 => 3 = 4 - 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 111 => 3 = 4 - 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 111 => 3 = 4 - 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 110 => 2 = 3 - 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 110 => 2 = 3 - 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 111 => 3 = 4 - 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 111 => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 111 => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1011 => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1101 => 2 = 3 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1101 => 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 1101 => 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1101 => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1110 => 3 = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 1010 => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1110 => 3 = 4 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => 1110 => 3 = 4 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 1110 => 3 = 4 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => 1110 => 3 = 4 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1110 => 3 = 4 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => 1100 => 2 = 3 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => 1100 => 2 = 3 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => 1110 => 3 = 4 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => 1110 => 3 = 4 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => 1110 => 3 = 4 - 1
[12,11,10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10,11,12] => [12,11,10,9,8,7,6,5,4,3,2,1] => 11111111111 => ? = 12 - 1
[7,8,9,10,11,12,1,2,3,4,5,6] => [1,2,3,4,5,6,7,8,9,10,11,12] => [12,11,10,9,8,7,6,5,4,3,2,1] => 11111111111 => ? = 12 - 1
[7,8,9,10,11,12,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10,11,12] => [12,11,10,9,8,7,6,5,4,3,2,1] => 11111111111 => ? = 12 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000725
(load all 3 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [1,2] => 2
[2,1] => [1,2] => [2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 3
[1,3,2] => [1,3,2] => [2,3,1] => [2,1,3] => 2
[2,1,3] => [1,3,2] => [2,3,1] => [2,1,3] => 2
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 3
[3,1,2] => [1,2,3] => [3,2,1] => [1,2,3] => 3
[3,2,1] => [1,2,3] => [3,2,1] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 3
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 3
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 3
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 3
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 3
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 3
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => 3
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => 4
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => 4
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [4,1,2,3,5] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 4
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => 4
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 3
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => 4
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => 4
[4,5,3,6,2,7,1,8] => [1,8,2,7,3,6,4,5] => [5,4,6,3,7,2,8,1] => [4,5,3,6,2,7,1,8] => ? = 5
[7,6,4,3,1,2,5,8] => [1,2,5,8,3,4,6,7] => [7,6,4,3,8,5,2,1] => [2,3,5,6,1,4,7,8] => ? = 6
[5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[4,5,2,3,1,6,7,8] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[5,2,3,4,1,6,7,8] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[2,3,4,5,1,6,7,8] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[1,6,7,8,5,4,3,2] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[2,1,6,7,8,5,4,3] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[3,2,1,6,7,8,5,4] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[4,3,2,1,6,7,8,5] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[3,4,2,1,6,7,8,5] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[2,3,4,1,6,7,8,5] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[2,4,3,6,5,1,8,7] => [1,8,2,4,3,6,5,7] => [7,5,6,3,4,2,8,1] => [2,4,3,6,5,7,1,8] => ? = 4
[4,3,5,2,6,1,8,7] => [1,8,2,6,3,5,4,7] => [7,4,5,3,6,2,8,1] => [2,5,4,6,3,7,1,8] => ? = 5
[3,5,4,2,7,6,1,8] => [1,8,2,7,3,5,4,6] => [6,4,5,3,7,2,8,1] => [3,5,4,6,2,7,1,8] => ? = 5
[4,2,3,1,6,7,8,5] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[5,2,4,3,6,1,8,7] => [1,8,2,4,3,6,5,7] => [7,5,6,3,4,2,8,1] => [2,4,3,6,5,7,1,8] => ? = 4
[2,4,1,6,3,8,5,7] => [1,6,2,4,3,8,5,7] => [7,5,8,3,4,2,6,1] => [2,4,1,6,5,7,3,8] => ? = 4
[2,4,1,6,3,7,5,8] => [1,6,2,4,3,7,5,8] => [8,5,7,3,4,2,6,1] => [1,4,2,6,5,7,3,8] => ? = 4
[2,4,1,5,3,7,6,8] => [1,5,2,4,3,7,6,8] => [8,6,7,3,4,2,5,1] => [1,3,2,6,5,7,4,8] => ? = 3
[2,4,1,5,3,8,6,7] => [1,5,2,4,3,8,6,7] => [7,6,8,3,4,2,5,1] => [2,3,1,6,5,7,4,8] => ? = 3
[2,6,1,7,3,8,4,5] => [1,7,2,6,3,8,4,5] => [5,4,8,3,6,2,7,1] => [4,5,1,6,3,7,2,8] => ? = 5
[2,1,6,7,8,3,4,5] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[2,3,1,6,7,8,4,5] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[3,4,6,7,1,2,5,8] => [1,2,5,8,3,4,6,7] => [7,6,4,3,8,5,2,1] => [2,3,5,6,1,4,7,8] => ? = 6
[3,5,1,7,2,8,4,6] => [1,7,2,8,3,5,4,6] => [6,4,5,3,8,2,7,1] => [3,5,4,6,1,7,2,8] => ? = 5
[1,3,2,5,4,8,6,7] => [1,3,2,5,4,8,6,7] => [7,6,8,4,5,2,3,1] => [2,3,1,5,4,7,6,8] => ? = 3
[3,6,1,7,2,8,4,5] => [1,7,2,8,3,6,4,5] => [5,4,6,3,8,2,7,1] => [4,5,3,6,1,7,2,8] => ? = 5
[1,3,2,6,4,8,5,7] => [1,3,2,6,4,8,5,7] => [7,5,8,4,6,2,3,1] => [2,4,1,5,3,7,6,8] => ? = 4
[1,3,2,7,4,8,5,6] => [1,3,2,7,4,8,5,6] => [6,5,8,4,7,2,3,1] => [3,4,1,5,2,7,6,8] => ? = 4
[4,1,5,2,6,3,8,7] => [1,5,2,6,3,8,4,7] => [7,4,8,3,6,2,5,1] => [2,5,1,6,3,7,4,8] => ? = 5
[4,5,1,6,2,8,3,7] => [1,6,2,8,3,7,4,5] => [5,4,7,3,8,2,6,1] => [4,5,2,6,1,7,3,8] => ? = 5
[4,5,1,6,2,7,3,8] => [1,6,2,7,3,8,4,5] => [5,4,8,3,7,2,6,1] => [4,5,1,6,2,7,3,8] => ? = 5
[4,1,5,2,7,3,8,6] => [1,5,2,7,3,8,4,6] => [6,4,8,3,7,2,5,1] => [3,5,1,6,2,7,4,8] => ? = 5
[4,5,1,7,2,8,3,6] => [1,7,2,8,3,6,4,5] => [5,4,6,3,8,2,7,1] => [4,5,3,6,1,7,2,8] => ? = 5
[1,4,2,5,3,8,6,7] => [1,4,2,5,3,8,6,7] => [7,6,8,3,5,2,4,1] => [2,3,1,6,4,7,5,8] => ? = 3
[4,1,6,2,7,3,8,5] => [1,6,2,7,3,8,4,5] => [5,4,8,3,7,2,6,1] => [4,5,1,6,2,7,3,8] => ? = 5
[1,4,2,6,3,8,5,7] => [1,4,2,6,3,8,5,7] => [7,5,8,3,6,2,4,1] => [2,4,1,6,3,7,5,8] => ? = 4
[1,4,2,7,3,8,5,6] => [1,4,2,7,3,8,5,6] => [6,5,8,3,7,2,4,1] => [3,4,1,6,2,7,5,8] => ? = 4
[5,1,6,2,7,3,8,4] => [1,6,2,7,3,8,4,5] => [5,4,8,3,7,2,6,1] => [4,5,1,6,2,7,3,8] => ? = 5
[1,5,2,6,3,8,4,7] => [1,5,2,6,3,8,4,7] => [7,4,8,3,6,2,5,1] => [2,5,1,6,3,7,4,8] => ? = 5
[1,5,2,7,3,8,4,6] => [1,5,2,7,3,8,4,6] => [6,4,8,3,7,2,5,1] => [3,5,1,6,2,7,4,8] => ? = 5
[1,6,2,7,3,8,4,5] => [1,6,2,7,3,8,4,5] => [5,4,8,3,7,2,6,1] => [4,5,1,6,2,7,3,8] => ? = 5
[1,6,7,8,2,3,4,5] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => [4,5,6,7,1,2,3,8] => ? = 7
[3,6,1,5,4,2,8,7] => [1,5,2,8,3,6,4,7] => [7,4,6,3,8,2,5,1] => [2,5,3,6,1,7,4,8] => ? = 5
[4,6,5,1,3,2,8,7] => [1,3,2,8,4,6,5,7] => [7,5,6,4,8,2,3,1] => [2,4,3,5,1,7,6,8] => ? = 4
[7,4,8,2,6,5,1,3] => [1,3,2,6,4,8,5,7] => [7,5,8,4,6,2,3,1] => [2,4,1,5,3,7,6,8] => ? = 4
[3,5,1,8,2,7,6,4] => [1,8,2,7,3,5,4,6] => [6,4,5,3,7,2,8,1] => [3,5,4,6,2,7,1,8] => ? = 5
[3,6,1,8,7,2,5,4] => [1,8,2,5,3,6,4,7] => [7,4,6,3,5,2,8,1] => [2,5,3,6,4,7,1,8] => ? = 5
[3,7,1,8,6,5,2,4] => [1,8,2,4,3,7,5,6] => [6,5,7,3,4,2,8,1] => [3,4,2,6,5,7,1,8] => ? = 4
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St001880
(load all 5 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 45%
Values
[1] => [1] => [1] => ([],1)
 => ? = 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 2
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 2
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => [1,3,2] => [3,1,2] => ([(1,2)],3)
 => ? = 2
[2,1,3] => [1,3,2] => [3,1,2] => ([(1,2)],3)
 => ? = 2
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[3,1,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[3,2,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? = 3
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? = 3
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 3
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? = 3
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? = 3
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? = 3
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? = 3
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? = 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ? = 3
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 3
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? = 3
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? = 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? = 4
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? = 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 3
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ? = 3
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4
[1,4,5,3,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 4
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
 => ? = 3
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
 => ? = 3
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 4
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 4
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 4
[2,1,3,4,5] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? = 4
[2,1,3,5,4] => [1,3,5,2,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? = 4
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[2,1,4,5,3] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 4
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 4
[2,3,1,4,5] => [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? = 4
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 4
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ? = 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[3,5,1,4,2] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[3,5,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,1,4,2,3] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[5,1,4,3,2] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[5,2,1,4,3] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[5,2,3,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,3,1,4,2] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[5,3,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 4
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 4
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 5
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 4
[1,4,3,5,6,2] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 5
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 4
[1,5,2,3,4,6] => [1,5,2,3,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
[1,5,3,4,6,2] => [1,5,2,3,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
[1,5,4,6,2,3] => [1,5,2,3,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001330
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 73%
Values
[1] => [1] => [1] => ([],1)
 => 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 2
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 2
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4
[1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,5,2,4,3] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,5,3,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,5,4,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,1,4,5,3] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4
[2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4
[2,3,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,3,5,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 6
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000454
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1] => ([],1)
 => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 2 - 1
[2,1,3] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 2 - 1
[2,3,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,1,2] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[3,2,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,5,4,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,3,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
[6,3,4,5,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(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)
 => 5 = 6 - 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001879
(load all 4 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 45%
Values
[1] => [1] => [1] => ([],1)
 => ? = 1 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
 => ? = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2] => [1,3,2] => [2,3,1] => ([(1,2)],3)
 => ? = 2 - 1
[2,1,3] => [1,3,2] => [2,3,1] => ([(1,2)],3)
 => ? = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,1,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 2 - 1
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[2,4,1,3] => [1,3,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 2 - 1
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 2 - 1
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3 - 1
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[4,1,3,2] => [1,3,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ? = 3 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? = 3 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ? = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ? = 4 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? = 4 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? = 3 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? = 3 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? = 4 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 3 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 3 - 1
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4 - 1
[1,5,4,2,3] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4 - 1
[1,5,4,3,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4 - 1
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ? = 4 - 1
[2,1,3,5,4] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4 - 1
[2,1,4,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? = 4 - 1
[2,1,4,5,3] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4 - 1
[2,1,5,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4 - 1
[2,1,5,4,3] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,3,4,5,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,3,4,5,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,4,5,1,2,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[6,4,5,2,3,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.