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Your data matches 165 different statistics following compositions of up to 3 maps.
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Matching statistic: St000456
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000290
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> 010 => 001 => 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> 010 => 001 => 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 010 => 001 => 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 0010 => 0001 => 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> 0010 => 0001 => 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0011 => 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> 0010 => 0001 => 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0011 => 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> 0010 => 0001 => 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0011 => 0 = 1 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> 00110 => 00011 => 0 = 1 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> 00110 => 00011 => 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> 01110 => 00111 => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> 00110 => 00011 => 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> 01110 => 00111 => 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> 00110 => 00011 => 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> 01110 => 00111 => 0 = 1 - 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> 010010 => 001001 => 3 = 4 - 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> 001010 => 000101 => 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> 010010 => 001001 => 3 = 4 - 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> 001010 => 000101 => 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> 000110 => 000011 => 0 = 1 - 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> 010010 => 001001 => 3 = 4 - 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> 001010 => 000101 => 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> 011010 => 001101 => 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> 010010 => 001001 => 3 = 4 - 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> 001010 => 000101 => 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> 010110 => 001011 => 3 = 4 - 1
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> 000110 => 000011 => 0 = 1 - 1
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000313
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000313: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000313: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The number of degree 2 vertices of a graph.
A vertex has degree 2 if and only if it lies on a unique maximal path.
Matching statistic: St001311
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The cyclomatic number of a graph.
This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph.
A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001319
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001319: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001319: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph.
A graph is a disjoint union of isolated vertices and a star if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001328
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001328: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.
A graph is bipartite if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001485
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001485: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St001485: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> 010 => 001 => 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> 010 => 001 => 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 010 => 001 => 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 0010 => 0001 => 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> 0010 => 0001 => 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0011 => 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> 0010 => 0001 => 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0011 => 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> 0010 => 0001 => 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0011 => 0 = 1 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> 00110 => 00011 => 0 = 1 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> 00110 => 00011 => 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> 01110 => 00111 => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> 00110 => 00011 => 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> 00010 => 00001 => 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> 01110 => 00111 => 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> 01010 => 00101 => 3 = 4 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> 00110 => 00011 => 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> 01110 => 00111 => 0 = 1 - 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> 010010 => 001001 => 3 = 4 - 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> 001010 => 000101 => 4 = 5 - 1
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> 010010 => 001001 => 3 = 4 - 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> 001010 => 000101 => 4 = 5 - 1
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> 000110 => 000011 => 0 = 1 - 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> 010010 => 001001 => 3 = 4 - 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> 001010 => 000101 => 4 = 5 - 1
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> 000010 => 000001 => 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> 011010 => 001101 => 4 = 5 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> 010010 => 001001 => 3 = 4 - 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> 001010 => 000101 => 4 = 5 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> 010110 => 001011 => 3 = 4 - 1
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> 000110 => 000011 => 0 = 1 - 1
Description
The modular major index of a binary word.
This is [[St000290]] modulo the length of the word.
Matching statistic: St001638
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001638: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 48%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001638: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 48%●distinct values known / distinct values provided: 33%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4,6],[3,7],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3,6],[4,7],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,2,3,4],[5,7],[6]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6],[2,5],[7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4,6],[3,5],[7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3,6],[4,5],[7]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4,6,7],[2],[3],[5]]
=> [[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6,7],[3],[4],[5]]
=> [[1,2,3,7],[4],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4,6],[3],[5],[7]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3,6],[4],[5],[7]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4,6],[2,5,7],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6],[2,5,7],[4]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6],[3,5,7],[4]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6],[3,4,7],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4],[2,5,7],[6]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4],[3,5,7],[6]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3],[4,5,7],[6]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4,6],[2,5],[3,7]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6],[2,5],[4,7]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6],[3,5],[4,7]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6],[2,4],[5,7]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6],[3,4],[5,7]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4],[2,5],[6,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4],[3,5],[6,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4,6],[2,7],[3],[5]]
=> [[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6],[3,7],[4],[5]]
=> [[1,2,3,7],[4],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4],[2,7],[5],[6]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,4],[3,7],[5],[6]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,2,3],[4,7],[5],[6]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4,6],[2,5],[3],[7]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6],[2,5],[4],[7]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6],[3,5],[4],[7]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6],[2,4],[5],[7]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6],[3,4],[5],[7]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,6,7],[2],[3],[4],[5]]
=> [[1,2,7],[3],[4],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4,6],[2],[3],[5],[7]]
=> [[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3,6],[2],[4],[5],[7]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2,6],[3],[4],[5],[7]]
=> [[1,2,3,7],[4],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,4],[2,5],[3,7],[6]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[1,2],[3,4],[5,7],[6]]
=> [[1,2,3,4],[5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,6],[2,7],[3],[4],[5]]
=> [[1,2,7],[3],[4],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,2],[3,7],[4],[5],[6]]
=> [[1,2,3,7],[4],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,6],[2],[3],[4],[5],[7]]
=> [[1,2,7],[3],[4],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
Description
The book thickness of a graph.
The book thickness (or pagenumber, or stacknumber) of a graph is the minimal number of pages required for a book embedding of a graph.
Matching statistic: St000455
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 48%●distinct values known / distinct values provided: 33%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 48%●distinct values known / distinct values provided: 33%
Values
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6,7],[3,4],[5]]
=> [[1,2,3,4],[5,7],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4,6],[3,7],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3,6],[4,7],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,3,4],[5,7],[6]]
=> [[1,2,3,4,5,7],[6]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2,5],[7]]
=> [[1,2,4,5,7],[3,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4,6],[3,5],[7]]
=> [[1,2,3,5,7],[4,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3,6],[4,5],[7]]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [[1,2,5,7],[3],[4],[6]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6,7],[3],[4],[5]]
=> [[1,2,3,7],[4],[5],[6]]
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,2,4,5,7],[3],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4,6],[3],[5],[7]]
=> [[1,2,3,5,7],[4],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3,6],[4],[5],[7]]
=> [[1,2,3,4,7],[5],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2,5,7],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6],[2,5,7],[4]]
=> [[1,2,4,5,7],[3],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6],[3,5,7],[4]]
=> [[1,2,3,5,7],[4],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6],[3,4,7],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4],[2,5,7],[6]]
=> [[1,2,4,5,7],[3,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4],[3,5,7],[6]]
=> [[1,2,3,5,7],[4,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3],[4,5,7],[6]]
=> [[1,2,3,4,5,7],[6]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2,5],[3,7]]
=> [[1,2,5,7],[3,6],[4]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6],[2,5],[4,7]]
=> [[1,2,4,5],[3,7],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6],[3,5],[4,7]]
=> [[1,2,3,5],[4,7],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6],[2,4],[5,7]]
=> [[1,2,4,7],[3,5],[6]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6],[3,4],[5,7]]
=> [[1,2,3,4],[5,7],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4],[2,5],[6,7]]
=> [[1,2,4,5,7],[3,6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4],[3,5],[6,7]]
=> [[1,2,3,5,7],[4,6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2,7],[3],[5]]
=> [[1,2,5,7],[3],[4],[6]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6],[3,7],[4],[5]]
=> [[1,2,3,7],[4],[5],[6]]
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,4],[2,7],[5],[6]]
=> [[1,2,4,5,7],[3],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,4],[3,7],[5],[6]]
=> [[1,2,3,5,7],[4],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,2,3],[4,7],[5],[6]]
=> [[1,2,3,4,7],[5],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2,5],[3],[7]]
=> [[1,2,5,7],[3,6],[4]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6],[2,5],[4],[7]]
=> [[1,2,4,5],[3,7],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6],[3,5],[4],[7]]
=> [[1,2,3,5],[4,7],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6],[2,4],[5],[7]]
=> [[1,2,4,7],[3,5],[6]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6],[3,4],[5],[7]]
=> [[1,2,3,4],[5,7],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,6,7],[2],[3],[4],[5]]
=> [[1,2,7],[3],[4],[5],[6]]
=> [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [[1,2,5,7],[3],[4],[6]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [[1,2,4,7],[3],[5],[6]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,6],[3],[4],[5],[7]]
=> [[1,2,3,7],[4],[5],[6]]
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4],[2,5],[3,7],[6]]
=> [[1,2,5,7],[3,6],[4]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 1
[[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,5],[3,7],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2],[3,4],[5,7],[6]]
=> [[1,2,3,4],[5,7],[6]]
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,6],[2,7],[3],[4],[5]]
=> [[1,2,7],[3],[4],[5],[6]]
=> [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2],[3,7],[4],[5],[6]]
=> [[1,2,3,7],[4],[5],[6]]
=> [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,6],[2],[3],[4],[5],[7]]
=> [[1,2,7],[3],[4],[5],[6]]
=> [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
The following 155 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St001644The dimension of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000095The number of triangles of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000741The Colin de Verdière graph invariant. St001260The permanent of an alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000787The number of flips required to make a perfect matching noncrossing. St001964The interval resolution global dimension of a poset. St000069The number of maximal elements of a poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001556The number of inversions of the third entry of a permutation. St001569The maximal modular displacement of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001344The neighbouring number of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001520The number of strict 3-descents. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001896The number of right descents of a signed permutations. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000542The number of left-to-right-minima of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001545The second Elser number of a connected graph. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001703The villainy of a graph. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001734The lettericity of a graph. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000545The number of parabolic double cosets with minimal element being the given permutation.
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