Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => => [1] => 1
[1,2] => [1,2] => 0 => [2] => 2
[2,1] => [1,2] => 0 => [2] => 2
[1,2,3] => [1,2,3] => 00 => [3] => 3
[1,3,2] => [1,3,2] => 01 => [2,1] => 2
[2,1,3] => [1,3,2] => 01 => [2,1] => 2
[2,3,1] => [1,2,3] => 00 => [3] => 3
[3,1,2] => [1,2,3] => 00 => [3] => 3
[3,2,1] => [1,2,3] => 00 => [3] => 3
[1,2,3,4] => [1,2,3,4] => 000 => [4] => 4
[1,2,4,3] => [1,2,4,3] => 001 => [3,1] => 3
[1,3,2,4] => [1,3,2,4] => 010 => [2,2] => 2
[1,3,4,2] => [1,3,4,2] => 001 => [3,1] => 3
[1,4,2,3] => [1,4,2,3] => 001 => [3,1] => 3
[1,4,3,2] => [1,4,2,3] => 001 => [3,1] => 3
[2,1,3,4] => [1,3,4,2] => 001 => [3,1] => 3
[2,1,4,3] => [1,4,2,3] => 001 => [3,1] => 3
[2,3,1,4] => [1,4,2,3] => 001 => [3,1] => 3
[2,3,4,1] => [1,2,3,4] => 000 => [4] => 4
[2,4,1,3] => [1,3,2,4] => 010 => [2,2] => 2
[2,4,3,1] => [1,2,4,3] => 001 => [3,1] => 3
[3,1,2,4] => [1,2,4,3] => 001 => [3,1] => 3
[3,1,4,2] => [1,4,2,3] => 001 => [3,1] => 3
[3,2,1,4] => [1,4,2,3] => 001 => [3,1] => 3
[3,2,4,1] => [1,2,4,3] => 001 => [3,1] => 3
[3,4,1,2] => [1,2,3,4] => 000 => [4] => 4
[3,4,2,1] => [1,2,3,4] => 000 => [4] => 4
[4,1,2,3] => [1,2,3,4] => 000 => [4] => 4
[4,1,3,2] => [1,3,2,4] => 010 => [2,2] => 2
[4,2,1,3] => [1,3,2,4] => 010 => [2,2] => 2
[4,2,3,1] => [1,2,3,4] => 000 => [4] => 4
[4,3,1,2] => [1,2,3,4] => 000 => [4] => 4
[4,3,2,1] => [1,2,3,4] => 000 => [4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => [5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => [4,1] => 4
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => [3,2] => 3
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => [4,1] => 4
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => [4,1] => 4
[1,2,5,4,3] => [1,2,5,3,4] => 0001 => [4,1] => 4
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => [2,3] => 2
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => [2,2,1] => 2
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => [3,2] => 3
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => [4,1] => 4
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => [4,1] => 4
[1,3,5,4,2] => [1,3,5,2,4] => 0001 => [4,1] => 4
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => [3,2] => 3
[1,4,2,5,3] => [1,4,2,5,3] => 0011 => [3,1,1] => 3
[1,4,3,2,5] => [1,4,2,5,3] => 0011 => [3,1,1] => 3
[1,4,3,5,2] => [1,4,2,3,5] => 0010 => [3,2] => 3
[1,4,5,2,3] => [1,4,5,2,3] => 0001 => [4,1] => 4
Description
The first part of an integer composition.
Matching statistic: St000326
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 100%●distinct values known / distinct values provided: 90%
Values
[1] => [1] => => ? = 1
[1,2] => [1,2] => 0 => 2
[2,1] => [1,2] => 0 => 2
[1,2,3] => [1,2,3] => 00 => 3
[1,3,2] => [1,3,2] => 01 => 2
[2,1,3] => [1,3,2] => 01 => 2
[2,3,1] => [1,2,3] => 00 => 3
[3,1,2] => [1,2,3] => 00 => 3
[3,2,1] => [1,2,3] => 00 => 3
[1,2,3,4] => [1,2,3,4] => 000 => 4
[1,2,4,3] => [1,2,4,3] => 001 => 3
[1,3,2,4] => [1,3,2,4] => 010 => 2
[1,3,4,2] => [1,3,4,2] => 001 => 3
[1,4,2,3] => [1,4,2,3] => 001 => 3
[1,4,3,2] => [1,4,2,3] => 001 => 3
[2,1,3,4] => [1,3,4,2] => 001 => 3
[2,1,4,3] => [1,4,2,3] => 001 => 3
[2,3,1,4] => [1,4,2,3] => 001 => 3
[2,3,4,1] => [1,2,3,4] => 000 => 4
[2,4,1,3] => [1,3,2,4] => 010 => 2
[2,4,3,1] => [1,2,4,3] => 001 => 3
[3,1,2,4] => [1,2,4,3] => 001 => 3
[3,1,4,2] => [1,4,2,3] => 001 => 3
[3,2,1,4] => [1,4,2,3] => 001 => 3
[3,2,4,1] => [1,2,4,3] => 001 => 3
[3,4,1,2] => [1,2,3,4] => 000 => 4
[3,4,2,1] => [1,2,3,4] => 000 => 4
[4,1,2,3] => [1,2,3,4] => 000 => 4
[4,1,3,2] => [1,3,2,4] => 010 => 2
[4,2,1,3] => [1,3,2,4] => 010 => 2
[4,2,3,1] => [1,2,3,4] => 000 => 4
[4,3,1,2] => [1,2,3,4] => 000 => 4
[4,3,2,1] => [1,2,3,4] => 000 => 4
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 5
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 3
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 4
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 4
[1,2,5,4,3] => [1,2,5,3,4] => 0001 => 4
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 2
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 3
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 4
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 4
[1,3,5,4,2] => [1,3,5,2,4] => 0001 => 4
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 3
[1,4,2,5,3] => [1,4,2,5,3] => 0011 => 3
[1,4,3,2,5] => [1,4,2,5,3] => 0011 => 3
[1,4,3,5,2] => [1,4,2,3,5] => 0010 => 3
[1,4,5,2,3] => [1,4,5,2,3] => 0001 => 4
[1,4,5,3,2] => [1,4,5,2,3] => 0001 => 4
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 100%●distinct values known / distinct values provided: 90%
Values
[1] => [1] => => => ? = 1 - 1
[1,2] => [1,2] => 0 => 1 => 1 = 2 - 1
[2,1] => [1,2] => 0 => 1 => 1 = 2 - 1
[1,2,3] => [1,2,3] => 00 => 11 => 2 = 3 - 1
[1,3,2] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[2,1,3] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[2,3,1] => [1,2,3] => 00 => 11 => 2 = 3 - 1
[3,1,2] => [1,2,3] => 00 => 11 => 2 = 3 - 1
[3,2,1] => [1,2,3] => 00 => 11 => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => 001 => 110 => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => 010 => 101 => 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => 001 => 110 => 2 = 3 - 1
[1,4,2,3] => [1,4,2,3] => 001 => 110 => 2 = 3 - 1
[1,4,3,2] => [1,4,2,3] => 001 => 110 => 2 = 3 - 1
[2,1,3,4] => [1,3,4,2] => 001 => 110 => 2 = 3 - 1
[2,1,4,3] => [1,4,2,3] => 001 => 110 => 2 = 3 - 1
[2,3,1,4] => [1,4,2,3] => 001 => 110 => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[2,4,1,3] => [1,3,2,4] => 010 => 101 => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => 001 => 110 => 2 = 3 - 1
[3,1,2,4] => [1,2,4,3] => 001 => 110 => 2 = 3 - 1
[3,1,4,2] => [1,4,2,3] => 001 => 110 => 2 = 3 - 1
[3,2,1,4] => [1,4,2,3] => 001 => 110 => 2 = 3 - 1
[3,2,4,1] => [1,2,4,3] => 001 => 110 => 2 = 3 - 1
[3,4,1,2] => [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[3,4,2,1] => [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[4,1,2,3] => [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[4,1,3,2] => [1,3,2,4] => 010 => 101 => 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => 010 => 101 => 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[4,3,1,2] => [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => 000 => 111 => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 1111 => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1110 => 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1101 => 2 = 3 - 1
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 1110 => 3 = 4 - 1
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 1110 => 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,3,4] => 0001 => 1110 => 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1011 => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 1101 => 2 = 3 - 1
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 1110 => 3 = 4 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 1110 => 3 = 4 - 1
[1,3,5,4,2] => [1,3,5,2,4] => 0001 => 1110 => 3 = 4 - 1
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 1101 => 2 = 3 - 1
[1,4,2,5,3] => [1,4,2,5,3] => 0011 => 1100 => 2 = 3 - 1
[1,4,3,2,5] => [1,4,2,5,3] => 0011 => 1100 => 2 = 3 - 1
[1,4,3,5,2] => [1,4,2,3,5] => 0010 => 1101 => 2 = 3 - 1
[1,4,5,2,3] => [1,4,5,2,3] => 0001 => 1110 => 3 = 4 - 1
[1,4,5,3,2] => [1,4,5,2,3] => 0001 => 1110 => 3 = 4 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000725
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
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] => [2,3,1] => [1,2,3] => 3
[1,3,2] => [1,3,2] => [3,2,1] => [2,1,3] => 2
[2,1,3] => [1,3,2] => [3,2,1] => [2,1,3] => 2
[2,3,1] => [1,2,3] => [2,3,1] => [1,2,3] => 3
[3,1,2] => [1,2,3] => [2,3,1] => [1,2,3] => 3
[3,2,1] => [1,2,3] => [2,3,1] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 3
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 3
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 4
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 3
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 3
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 4
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 4
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 4
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 2
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 4
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 4
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [1,3,4,2,5] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => [1,4,2,3,5] => 4
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => [1,4,2,3,5] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => [2,1,3,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => [2,1,4,3,5] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => [2,3,1,4,5] => 3
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,1,5] => 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] => [3,4,2,5,1] => [3,1,2,4,5] => 3
[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] => [3,4,2,5,1] => [3,1,2,4,5] => 3
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => [3,4,1,2,5] => 4
[1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 5
[1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 5
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [2,4,6,5,3,7,1] => [1,5,2,4,3,6,7] => ? = 4
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [2,4,7,5,3,6,1] => [1,5,2,4,6,3,7] => ? = 5
[1,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 5
[1,2,6,3,7,5,4] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 5
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [2,4,7,5,3,6,1] => [1,5,2,4,6,3,7] => ? = 5
[1,2,6,4,3,7,5] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 5
[1,2,6,4,5,3,7] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 5
[1,2,6,4,5,7,3] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 5
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [2,4,6,5,3,7,1] => [1,5,2,4,3,6,7] => ? = 4
[1,2,6,4,7,5,3] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 5
[1,2,6,5,3,4,7] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 5
[1,2,6,5,3,7,4] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 5
[1,2,6,5,4,3,7] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 5
[1,2,6,5,4,7,3] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 5
[1,2,6,5,7,3,4] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 5
[1,2,6,5,7,4,3] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 5
[1,2,6,7,3,4,5] => [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 6
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [2,5,7,6,3,4,1] => [1,5,6,2,4,3,7] => ? = 4
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [2,5,7,6,3,4,1] => [1,5,6,2,4,3,7] => ? = 4
[1,2,6,7,4,5,3] => [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 6
[1,2,6,7,5,3,4] => [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 6
[1,2,6,7,5,4,3] => [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 6
[1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 5
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 4
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [2,4,7,5,6,3,1] => [1,6,2,4,5,3,7] => ? = 5
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 5
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 5
[1,2,7,4,3,5,6] => [1,2,7,3,5,6,4] => [2,4,7,5,6,3,1] => [1,6,2,4,5,3,7] => ? = 5
[1,2,7,4,3,6,5] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 5
[1,2,7,4,5,3,6] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 5
[1,2,7,4,5,6,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,4,6,3,5] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 4
[1,2,7,4,6,5,3] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 5
[1,2,7,5,3,4,6] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 5
[1,2,7,5,3,6,4] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 5
[1,2,7,5,4,3,6] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 5
[1,2,7,5,4,6,3] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 5
[1,2,7,5,6,3,4] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,5,6,4,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,6,3,4,5] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,6,3,5,4] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 4
[1,2,7,6,4,3,5] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 4
[1,2,7,6,4,5,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,6,5,3,4] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,2,7,6,5,4,3] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 6
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [3,2,4,5,7,6,1] => [2,1,3,4,6,5,7] => ? = 2
[1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [3,2,4,6,5,7,1] => [2,1,3,5,4,6,7] => ? = 2
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St001330
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 80%
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)
 => ? = 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[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)
 => ? = 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
[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)
 => ? = 3
[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)
 => ? = 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
[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)
 => ? = 4
[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)
 => ? = 4
[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)
 => ? = 4
[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)
 => ? = 4
[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)
 => ? = 2
[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)
 => ? = 3
[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)
 => ? = 3
[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)
 => ? = 3
[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)
 => ? = 3
[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)
 => ? = 3
[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)
 => ? = 4
[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: St001880
(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
St001880: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
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] => [2,3,1] => ([(1,2)],3)
 => ? = 2
[2,1,3] => [1,3,2] => [2,3,1] => ([(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] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 3
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ? = 3
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ? = 3
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],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] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 3
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 3
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? = 3
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 3
[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] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[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] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 4
[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
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ? = 4
[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)
 => ? = 4
[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)
 => ? = 4
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[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)
 => ? = 3
[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
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? = 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? = 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? = 3
[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)
 => ? = 3
[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
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? = 4
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 3
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4
[1,5,4,2,3] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4
[1,5,4,3,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? = 4
[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
[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
[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)
 => ? = 3
[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
[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
[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
[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
[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
[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
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5,3,4,1,2] => [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
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
[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)
 => 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
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)
 => ? = 3 - 1
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 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)
 => ? = 2 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 3 - 1
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 3 - 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)
 => ? = 3 - 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)
 => ? = 2 - 1
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 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)
 => ? = 4 - 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)
 => ? = 4 - 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)
 => ? = 4 - 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)
 => ? = 4 - 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)
 => ? = 2 - 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)
 => ? = 3 - 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)
 => ? = 3 - 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)
 => ? = 3 - 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)
 => ? = 3 - 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.