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Your data matches 235 different statistics following compositions of up to 3 maps.
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Matching statistic: St000758
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 2
[2,1] => 1
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 2
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 2
[1,1,3] => 2
[1,2,1,1] => 2
[1,3,1] => 2
[1,4] => 2
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 1
[3,2] => 2
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 2
[1,1,1,3] => 2
[1,1,2,1,1] => 2
[1,1,3,1] => 2
[1,1,4] => 2
[1,2,1,1,1] => 2
[1,3,1,1] => 2
[1,4,1] => 2
[1,5] => 2
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 2
[2,1,3] => 2
[2,2,1,1] => 2
[2,3,1] => 2
[2,4] => 2
[3,1,1,1] => 1
[3,1,2] => 2
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000093
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2] => ([],2)
=> ([],1)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],1)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4] => ([],4)
=> ([],1)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[5] => ([],5)
=> ([],1)
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(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)
=> 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(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)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[3,1,1,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000390
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 1
[1,1] => [2] => 10 => 1
[2] => [1,1] => 11 => 1
[1,1,1] => [3] => 100 => 1
[1,2] => [2,1] => 101 => 2
[2,1] => [1,2] => 110 => 1
[3] => [1,1,1] => 111 => 1
[1,1,1,1] => [4] => 1000 => 1
[1,1,2] => [3,1] => 1001 => 2
[1,2,1] => [2,2] => 1010 => 2
[1,3] => [2,1,1] => 1011 => 2
[2,1,1] => [1,3] => 1100 => 1
[2,2] => [1,2,1] => 1101 => 2
[3,1] => [1,1,2] => 1110 => 1
[4] => [1,1,1,1] => 1111 => 1
[1,1,1,1,1] => [5] => 10000 => 1
[1,1,1,2] => [4,1] => 10001 => 2
[1,1,2,1] => [3,2] => 10010 => 2
[1,1,3] => [3,1,1] => 10011 => 2
[1,2,1,1] => [2,3] => 10100 => 2
[1,3,1] => [2,1,2] => 10110 => 2
[1,4] => [2,1,1,1] => 10111 => 2
[2,1,1,1] => [1,4] => 11000 => 1
[2,1,2] => [1,3,1] => 11001 => 2
[2,2,1] => [1,2,2] => 11010 => 2
[2,3] => [1,2,1,1] => 11011 => 2
[3,1,1] => [1,1,3] => 11100 => 1
[3,2] => [1,1,2,1] => 11101 => 2
[4,1] => [1,1,1,2] => 11110 => 1
[5] => [1,1,1,1,1] => 11111 => 1
[1,1,1,1,1,1] => [6] => 100000 => 1
[1,1,1,1,2] => [5,1] => 100001 => 2
[1,1,1,2,1] => [4,2] => 100010 => 2
[1,1,1,3] => [4,1,1] => 100011 => 2
[1,1,2,1,1] => [3,3] => 100100 => 2
[1,1,3,1] => [3,1,2] => 100110 => 2
[1,1,4] => [3,1,1,1] => 100111 => 2
[1,2,1,1,1] => [2,4] => 101000 => 2
[1,3,1,1] => [2,1,3] => 101100 => 2
[1,4,1] => [2,1,1,2] => 101110 => 2
[1,5] => [2,1,1,1,1] => 101111 => 2
[2,1,1,1,1] => [1,5] => 110000 => 1
[2,1,1,2] => [1,4,1] => 110001 => 2
[2,1,2,1] => [1,3,2] => 110010 => 2
[2,1,3] => [1,3,1,1] => 110011 => 2
[2,2,1,1] => [1,2,3] => 110100 => 2
[2,3,1] => [1,2,1,2] => 110110 => 2
[2,4] => [1,2,1,1,1] => 110111 => 2
[3,1,1,1] => [1,1,4] => 111000 => 1
[3,1,2] => [1,1,3,1] => 111001 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000786
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2] => ([],2)
=> ([],1)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],1)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4] => ([],4)
=> ([],1)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[5] => ([],5)
=> ([],1)
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(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)
=> 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(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)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[3,1,1,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
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: St001337
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2] => ([],2)
=> ([],1)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],1)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4] => ([],4)
=> ([],1)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[5] => ([],5)
=> ([],1)
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(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)
=> 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(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)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[3,1,1,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2] => ([],2)
=> ([],1)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],1)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4] => ([],4)
=> ([],1)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[5] => ([],5)
=> ([],1)
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(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)
=> 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(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)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
[3,1,1,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St000292
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 0 = 1 - 1
[1,1] => [2] => 10 => 0 = 1 - 1
[2] => [1,1] => 11 => 0 = 1 - 1
[1,1,1] => [3] => 100 => 0 = 1 - 1
[1,2] => [2,1] => 101 => 1 = 2 - 1
[2,1] => [1,2] => 110 => 0 = 1 - 1
[3] => [1,1,1] => 111 => 0 = 1 - 1
[1,1,1,1] => [4] => 1000 => 0 = 1 - 1
[1,1,2] => [3,1] => 1001 => 1 = 2 - 1
[1,2,1] => [2,2] => 1010 => 1 = 2 - 1
[1,3] => [2,1,1] => 1011 => 1 = 2 - 1
[2,1,1] => [1,3] => 1100 => 0 = 1 - 1
[2,2] => [1,2,1] => 1101 => 1 = 2 - 1
[3,1] => [1,1,2] => 1110 => 0 = 1 - 1
[4] => [1,1,1,1] => 1111 => 0 = 1 - 1
[1,1,1,1,1] => [5] => 10000 => 0 = 1 - 1
[1,1,1,2] => [4,1] => 10001 => 1 = 2 - 1
[1,1,2,1] => [3,2] => 10010 => 1 = 2 - 1
[1,1,3] => [3,1,1] => 10011 => 1 = 2 - 1
[1,2,1,1] => [2,3] => 10100 => 1 = 2 - 1
[1,3,1] => [2,1,2] => 10110 => 1 = 2 - 1
[1,4] => [2,1,1,1] => 10111 => 1 = 2 - 1
[2,1,1,1] => [1,4] => 11000 => 0 = 1 - 1
[2,1,2] => [1,3,1] => 11001 => 1 = 2 - 1
[2,2,1] => [1,2,2] => 11010 => 1 = 2 - 1
[2,3] => [1,2,1,1] => 11011 => 1 = 2 - 1
[3,1,1] => [1,1,3] => 11100 => 0 = 1 - 1
[3,2] => [1,1,2,1] => 11101 => 1 = 2 - 1
[4,1] => [1,1,1,2] => 11110 => 0 = 1 - 1
[5] => [1,1,1,1,1] => 11111 => 0 = 1 - 1
[1,1,1,1,1,1] => [6] => 100000 => 0 = 1 - 1
[1,1,1,1,2] => [5,1] => 100001 => 1 = 2 - 1
[1,1,1,2,1] => [4,2] => 100010 => 1 = 2 - 1
[1,1,1,3] => [4,1,1] => 100011 => 1 = 2 - 1
[1,1,2,1,1] => [3,3] => 100100 => 1 = 2 - 1
[1,1,3,1] => [3,1,2] => 100110 => 1 = 2 - 1
[1,1,4] => [3,1,1,1] => 100111 => 1 = 2 - 1
[1,2,1,1,1] => [2,4] => 101000 => 1 = 2 - 1
[1,3,1,1] => [2,1,3] => 101100 => 1 = 2 - 1
[1,4,1] => [2,1,1,2] => 101110 => 1 = 2 - 1
[1,5] => [2,1,1,1,1] => 101111 => 1 = 2 - 1
[2,1,1,1,1] => [1,5] => 110000 => 0 = 1 - 1
[2,1,1,2] => [1,4,1] => 110001 => 1 = 2 - 1
[2,1,2,1] => [1,3,2] => 110010 => 1 = 2 - 1
[2,1,3] => [1,3,1,1] => 110011 => 1 = 2 - 1
[2,2,1,1] => [1,2,3] => 110100 => 1 = 2 - 1
[2,3,1] => [1,2,1,2] => 110110 => 1 = 2 - 1
[2,4] => [1,2,1,1,1] => 110111 => 1 = 2 - 1
[3,1,1,1] => [1,1,4] => 111000 => 0 = 1 - 1
[3,1,2] => [1,1,3,1] => 111001 => 1 = 2 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000386
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000068
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [.,.]
=> ([],1)
=> 1
[1,1] => [1,0,1,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> 1
[2] => [1,1,0,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,2] => [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1] => [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[3] => [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[.,.],[.,[.,.]]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],[.,[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
Description
The number of minimal elements in a poset.
Matching statistic: St000071
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [.,.]
=> ([],1)
=> 1
[1,1] => [1,0,1,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> 1
[2] => [1,1,0,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,2] => [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1] => [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[3] => [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[.,.],[.,[.,.]]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],[.,[.,[.,.]]]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
Description
The number of maximal chains in a poset.
The following 225 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000172The Grundy number of a graph. St000183The side length of the Durfee square of an integer partition. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000527The width of the poset. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000679The pruning number of an ordered tree. St000920The logarithmic height of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001029The size of the core of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001261The Castelnuovo-Mumford regularity of a graph. St001280The number of parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001657The number of twos in an integer partition. St001670The connected partition number of a graph. St001716The 1-improper chromatic number of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001963The tree-depth of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000159The number of distinct parts of the integer partition. St000272The treewidth of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000552The number of cut vertices of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000897The number of different multiplicities of parts of an integer partition. St000935The number of ordered refinements of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001175The size of a partition minus the hook length of the base cell. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001389The number of partitions of the same length below the given integer partition. St001393The induced matching number of a graph. St001459The number of zero columns in the nullspace of a graph. St001689The number of celebrities in a graph. St001712The number of natural descents of a standard Young tableau. St001743The discrepancy of a graph. St001792The arboricity of a graph. St000069The number of maximal elements of a poset. St000201The number of leaf nodes in a binary tree. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000257The number of distinct parts of a partition that occur at least twice. St000845The maximal number of elements covered by an element in a poset. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000453The number of distinct Laplacian eigenvalues of a graph. St000522The number of 1-protected nodes of a rooted tree. St000568The hook number of a binary tree. St000822The Hadwiger number of the graph. St001093The detour number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000204The number of internal nodes of a binary tree. St000632The jump number of the poset. St000846The maximal number of elements covering an element of a poset. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001512The minimum rank of a graph. St001812The biclique partition number of a graph. St001840The number of descents of a set partition. St000080The rank of the poset. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000565The major index of a set partition. St000480The number of lower covers of a partition in dominance order. St000053The number of valleys of the Dyck path. St000526The number of posets with combinatorially isomorphic order polytopes. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000640The rank of the largest boolean interval in a poset. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to 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 Dyck path as follows:
St001592The maximal number of simple paths between any two different vertices of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000306The bounce count of a Dyck path. St000647The number of big descents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001083The number of boxed occurrences of 132 in a permutation. St000353The number of inner valleys of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000862The number of parts of the shifted shape of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000356The number of occurrences of the pattern 13-2. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000779The tier of a permutation. St000919The number of maximal left branches of a binary tree. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000149The number of cells of the partition whose leg is zero and arm is odd. St000486The number of cycles of length at least 3 of a permutation. St001668The number of points of the poset minus the width of the poset. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001735The number of permutations with the same set of runs. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000619The number of cyclic descents of a permutation. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000360The number of occurrences of the pattern 32-1. St000781The number of proper colouring schemes of a Ferrers diagram. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000455The second largest eigenvalue of a graph if it is integral. St001586The number of odd parts smaller than the largest even part in an integer partition. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000015The number of peaks of a Dyck path. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000659The number of rises of length at least 2 of a Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001427The number of descents of a signed permutation. St000358The number of occurrences of the pattern 31-2. St000397The Strahler number of a rooted tree. St000523The number of 2-protected nodes of a rooted tree. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in 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 $K[x]/(x^n)$. St000742The number of big ascents of a permutation after prepending zero. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000035The number of left outer peaks of a permutation. St000662The staircase size of the code of a permutation. St000871The number of very big ascents of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000264The girth of a graph, which is not a tree. St001394The genus of a permutation. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000456The monochromatic index of a connected graph. St000470The number of runs in a permutation. St000245The number of ascents of a permutation. St001571The Cartan determinant of the integer partition. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001487The number of inner corners of a skew partition. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000991The number of right-to-left minima of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St000039The number of crossings of a permutation. St000252The number of nodes of degree 3 of a binary tree. 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 $eA$. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001435The number of missing boxes in the first row. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St000761The number of ascents in an integer composition. St000805The number of peaks of the associated bargraph. St000807The sum of the heights of the valleys of the associated bargraph. St001651The Frankl number of a lattice.
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