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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000784
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St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 3
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 3
[3,1,1]
=> 3
[2,2,1]
=> 3
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 4
[4,1,1]
=> 4
[3,3]
=> 3
[3,2,1]
=> 3
[3,1,1,1]
=> 4
[2,2,2]
=> 3
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> 6
[5,2]
=> 5
[4,3]
=> 4
[4,2,1]
=> 4
[3,3,1]
=> 3
[3,2,2]
=> 3
[3,2,1,1]
=> 4
[2,2,2,1]
=> 4
[2,2,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
[5,3]
=> 5
[4,4]
=> 4
[4,3,1]
=> 4
[4,2,2]
=> 4
[3,3,2]
=> 3
[3,3,1,1]
=> 4
[3,2,2,1]
=> 4
[2,2,2,2]
=> 4
[2,2,2,1,1]
=> 5
[5,4]
=> 5
[4,4,1]
=> 4
[4,3,2]
=> 4
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St001417
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Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001417: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
St001417: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 2
[1,1]
=> 110 => 011 => 2
[3]
=> 1000 => 0001 => 3
[2,1]
=> 1010 => 0011 => 2
[1,1,1]
=> 1110 => 0111 => 3
[4]
=> 10000 => 00001 => 4
[3,1]
=> 10010 => 00011 => 3
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 00111 => 3
[1,1,1,1]
=> 11110 => 01111 => 4
[5]
=> 100000 => 000001 => 5
[4,1]
=> 100010 => 000011 => 4
[3,2]
=> 10100 => 00011 => 3
[3,1,1]
=> 100110 => 000111 => 3
[2,2,1]
=> 11010 => 00111 => 3
[2,1,1,1]
=> 101110 => 001111 => 4
[1,1,1,1,1]
=> 111110 => 011111 => 5
[6]
=> 1000000 => 0000001 => 6
[5,1]
=> 1000010 => 0000011 => 5
[4,2]
=> 100100 => 000011 => 4
[4,1,1]
=> 1000110 => 0000111 => 4
[3,3]
=> 11000 => 00011 => 3
[3,2,1]
=> 101010 => 001011 => 3
[3,1,1,1]
=> 1001110 => 0001111 => 4
[2,2,2]
=> 11100 => 00111 => 3
[2,2,1,1]
=> 110110 => 001111 => 4
[2,1,1,1,1]
=> 1011110 => 0011111 => 5
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 6
[5,2]
=> 1000100 => 0000011 => 5
[4,3]
=> 101000 => 000011 => 4
[4,2,1]
=> 1001010 => 0001011 => 4
[3,3,1]
=> 110010 => 000111 => 3
[3,2,2]
=> 101100 => 000111 => 3
[3,2,1,1]
=> 1010110 => 0010111 => 4
[2,2,2,1]
=> 111010 => 001111 => 4
[2,2,1,1,1]
=> 1101110 => 0011111 => 5
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 7
[5,3]
=> 1001000 => 0000011 => 5
[4,4]
=> 110000 => 000011 => 4
[4,3,1]
=> 1010010 => 0001011 => 4
[4,2,2]
=> 1001100 => 0000111 => 4
[3,3,2]
=> 110100 => 000111 => 3
[3,3,1,1]
=> 1100110 => 0001111 => 4
[3,2,2,1]
=> 1011010 => 0010111 => 4
[2,2,2,2]
=> 111100 => 001111 => 4
[2,2,2,1,1]
=> 1110110 => 0011111 => 5
[5,4]
=> 1010000 => 0000011 => 5
[4,4,1]
=> 1100010 => 0000111 => 4
[4,3,2]
=> 1010100 => 0001011 => 4
Description
The length of a longest palindromic subword of a binary word.
A subword of a word is a word obtained by deleting letters. This statistic records the maximal length of a palindromic subword.
Any binary word of length $n$ contains a palindromic subword of length at least $n/2$, obtained by removing all occurrences of the letter which occurs less often. This bound is obtained by the word beginning with $n/2$ zeros and ending with $n/2$ ones.
Matching statistic: St000786
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [[]]
=> ([(0,1)],2)
=> 1
[2]
=> [1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[1,1]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 2
[3]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,2]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[],[[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[],[],[[[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[],[[[]],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 7
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[],[],[[[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 5
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[],[[[],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[[[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[[[],[]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 4
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 4
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 5
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[],[[[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 5
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[[[],[],[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 4
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 4
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St000808
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => [1,1] => 1
[2]
=> 100 => 001 => [2,1] => 2
[1,1]
=> 110 => 011 => [1,2] => 2
[3]
=> 1000 => 0001 => [3,1] => 3
[2,1]
=> 1010 => 0011 => [2,2] => 2
[1,1,1]
=> 1110 => 0111 => [1,3] => 3
[4]
=> 10000 => 00001 => [4,1] => 4
[3,1]
=> 10010 => 00011 => [3,2] => 3
[2,2]
=> 1100 => 0011 => [2,2] => 2
[2,1,1]
=> 10110 => 00111 => [2,3] => 3
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 4
[5]
=> 100000 => 000001 => [5,1] => 5
[4,1]
=> 100010 => 000011 => [4,2] => 4
[3,2]
=> 10100 => 00011 => [3,2] => 3
[3,1,1]
=> 100110 => 000111 => [3,3] => 3
[2,2,1]
=> 11010 => 00111 => [2,3] => 3
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 4
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 5
[6]
=> 1000000 => 0000001 => [6,1] => 6
[5,1]
=> 1000010 => 0000011 => [5,2] => 5
[4,2]
=> 100100 => 000011 => [4,2] => 4
[4,1,1]
=> 1000110 => 0000111 => [4,3] => 4
[3,3]
=> 11000 => 00011 => [3,2] => 3
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 3
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 4
[2,2,2]
=> 11100 => 00111 => [2,3] => 3
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 4
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 5
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 6
[5,2]
=> 1000100 => 0000011 => [5,2] => 5
[4,3]
=> 101000 => 000011 => [4,2] => 4
[4,2,1]
=> 1001010 => 0001011 => [3,1,1,2] => 4
[3,3,1]
=> 110010 => 000111 => [3,3] => 3
[3,2,2]
=> 101100 => 000111 => [3,3] => 3
[3,2,1,1]
=> 1010110 => 0010111 => [2,1,1,3] => 4
[2,2,2,1]
=> 111010 => 001111 => [2,4] => 4
[2,2,1,1,1]
=> 1101110 => 0011111 => [2,5] => 5
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => [1,7] => 7
[5,3]
=> 1001000 => 0000011 => [5,2] => 5
[4,4]
=> 110000 => 000011 => [4,2] => 4
[4,3,1]
=> 1010010 => 0001011 => [3,1,1,2] => 4
[4,2,2]
=> 1001100 => 0000111 => [4,3] => 4
[3,3,2]
=> 110100 => 000111 => [3,3] => 3
[3,3,1,1]
=> 1100110 => 0001111 => [3,4] => 4
[3,2,2,1]
=> 1011010 => 0010111 => [2,1,1,3] => 4
[2,2,2,2]
=> 111100 => 001111 => [2,4] => 4
[2,2,2,1,1]
=> 1110110 => 0011111 => [2,5] => 5
[5,4]
=> 1010000 => 0000011 => [5,2] => 5
[4,4,1]
=> 1100010 => 0000111 => [4,3] => 4
[4,3,2]
=> 1010100 => 0001011 => [3,1,1,2] => 4
Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
Matching statistic: St001566
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 4
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 4
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,2,4,1,5,6] => 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 7
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,4,3,6,2,1] => 5
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 4
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 4
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 3
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,2,1,5,6] => 4
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,2,4,5,1,6] => 4
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => 5
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => 5
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 4
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,3,5,2,6,1] => 4
Description
The length of the longest arithmetic progression in a permutation.
For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with
$$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
Matching statistic: St000923
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 97%●distinct values known / distinct values provided: 71%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 97%●distinct values known / distinct values provided: 71%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => ? = 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 4
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 4
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 4
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,2,4] => 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => ? = 7
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => 5
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 4
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 4
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => 4
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,2,5] => 4
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 4
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => 5
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 5
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 4
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => 4
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 3
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => 4
Description
The minimal number with no two order isomorphic substrings of this length in a permutation.
For example, the length $3$ substrings of the permutation $12435$ are $124$, $243$ and $435$, whereas its length $2$ substrings are $12$, $24$, $43$ and $35$.
No two sequences among $124$, $243$ and $435$ are order isomorphic, but $12$ and $24$ are, so the statistic on $12435$ is $3$.
This is inspired by [[St000922]].
Matching statistic: St000141
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,3,1] => 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 3 = 4 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,4,1] => 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 3 = 4 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 4 = 5 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,5,4,3,1] => 3 = 4 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,4,3,2,1,6] => 4 = 5 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => 5 = 6 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,6,5,4,3,1] => 4 = 5 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => 3 = 4 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => 3 = 4 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,5,4,3] => 2 = 3 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => 2 = 3 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 3 = 4 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 2 = 3 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [5,4,3,2,6,1] => 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => 5 = 6 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => 4 = 5 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,5,1,4,3] => 3 = 4 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,5,4,1,2] => 3 = 4 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2,5,1,4] => 2 = 3 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => 3 = 4 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [5,4,3,1,6,2] => 4 = 5 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => 6 = 7 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [2,6,5,1,4,3] => 4 = 5 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => 3 = 4 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 3 = 4 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 3 = 4 - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2,5,4,1] => 2 = 3 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 3 = 4 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3 = 4 - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [4,3,2,1,6,5] => 3 = 4 - 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [5,4,2,1,6,3] => 4 = 5 - 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,1,6,2,3,4] => [2,6,1,5,4,3] => 4 = 5 - 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,6,1,2,3,5] => [3,1,6,5,4,2] => 3 = 4 - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,5,2,4,1] => 3 = 4 - 1
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? = 4 - 1
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => [4,3,2,1,7,6,5] => ? = 4 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000956
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 89%●distinct values known / distinct values provided: 71%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 89%●distinct values known / distinct values provided: 71%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 5
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 7
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 5
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 4
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 4
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 4
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 4
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => 5
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => 5
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => 4
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 4
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 3
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => 4
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 5
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ? = 5
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 4
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 4
Description
The maximal displacement of a permutation.
This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$.
This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St001655
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001655: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001655: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1
[2]
=> [[1,2]]
=> [1,2] => ([],2)
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([],4)
=> 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([],5)
=> 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,1,1]
=> [[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
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([],6)
=> 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,1,1,1]
=> [[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
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 4
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 4
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 4
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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)
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 5
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 4
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
=> ? = 5
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 4
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 4
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? = 3
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 4
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 5
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9)],10)
=> ? = 5
[4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [9,10,5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [8,9,10,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9)],10)
=> ? = 4
[3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [10,7,8,9,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 4
[3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [9,10,7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 5
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 4
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ?
=> ? = 4
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ?
=> ? = 4
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ?
=> ? = 4
Description
The general position number of a graph.
A set $S$ of vertices in a graph $G$ is a general position set if no three vertices of $S$ lie on a shortest path between any two of them.
Matching statistic: St001656
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1
[2]
=> [[1,2]]
=> [1,2] => ([],2)
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([],4)
=> 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([],5)
=> 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,1,1]
=> [[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
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([],6)
=> 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,1,1,1]
=> [[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
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 4
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 4
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 4
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(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)
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 5
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 4
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)],9)
=> ? = 5
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 4
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 4
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? = 3
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? = 4
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 5
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9)],10)
=> ? = 5
[4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [9,10,5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [8,9,10,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9)],10)
=> ? = 4
[3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [10,7,8,9,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 4
[3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [9,10,7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 5
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 4
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ?
=> ? = 4
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ?
=> ? = 4
[3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ?
=> ? = 4
Description
The monophonic position number of a graph.
A subset $M$ of the vertex set of a graph is a monophonic position set if no three vertices of $M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001879The 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000834The number of right outer peaks of a permutation. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000075The orbit size of a standard tableau under promotion. St000099The number of valleys of a permutation, including the boundary. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001645The pebbling number of a connected graph. St000023The number of inner peaks of a permutation. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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