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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
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(load all 12 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 => 0 = 1 - 1
[2]
=> 0 => 1 => 1 = 2 - 1
[1,1]
=> 11 => 00 => 0 = 1 - 1
[3]
=> 1 => 0 => 0 = 1 - 1
[2,1]
=> 01 => 10 => 1 = 2 - 1
[1,1,1]
=> 111 => 000 => 0 = 1 - 1
[4]
=> 0 => 1 => 1 = 2 - 1
[3,1]
=> 11 => 00 => 0 = 1 - 1
[2,2]
=> 00 => 11 => 2 = 3 - 1
[2,1,1]
=> 011 => 100 => 1 = 2 - 1
[1,1,1,1]
=> 1111 => 0000 => 0 = 1 - 1
[5]
=> 1 => 0 => 0 = 1 - 1
[4,1]
=> 01 => 10 => 1 = 2 - 1
[3,2]
=> 10 => 01 => 1 = 2 - 1
[3,1,1]
=> 111 => 000 => 0 = 1 - 1
[2,2,1]
=> 001 => 110 => 2 = 3 - 1
[2,1,1,1]
=> 0111 => 1000 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 00000 => 0 = 1 - 1
[6]
=> 0 => 1 => 1 = 2 - 1
[5,1]
=> 11 => 00 => 0 = 1 - 1
[4,2]
=> 00 => 11 => 2 = 3 - 1
[4,1,1]
=> 011 => 100 => 1 = 2 - 1
[3,3]
=> 11 => 00 => 0 = 1 - 1
[3,2,1]
=> 101 => 010 => 1 = 2 - 1
[3,1,1,1]
=> 1111 => 0000 => 0 = 1 - 1
[2,2,2]
=> 000 => 111 => 3 = 4 - 1
[2,2,1,1]
=> 0011 => 1100 => 2 = 3 - 1
[2,1,1,1,1]
=> 01111 => 10000 => 1 = 2 - 1
[7]
=> 1 => 0 => 0 = 1 - 1
[6,1]
=> 01 => 10 => 1 = 2 - 1
[5,2]
=> 10 => 01 => 1 = 2 - 1
[5,1,1]
=> 111 => 000 => 0 = 1 - 1
[4,3]
=> 01 => 10 => 1 = 2 - 1
[4,2,1]
=> 001 => 110 => 2 = 3 - 1
[4,1,1,1]
=> 0111 => 1000 => 1 = 2 - 1
[3,3,1]
=> 111 => 000 => 0 = 1 - 1
[3,2,2]
=> 100 => 011 => 2 = 3 - 1
[3,2,1,1]
=> 1011 => 0100 => 1 = 2 - 1
[3,1,1,1,1]
=> 11111 => 00000 => 0 = 1 - 1
[2,2,2,1]
=> 0001 => 1110 => 3 = 4 - 1
[2,2,1,1,1]
=> 00111 => 11000 => 2 = 3 - 1
[8]
=> 0 => 1 => 1 = 2 - 1
[7,1]
=> 11 => 00 => 0 = 1 - 1
[6,2]
=> 00 => 11 => 2 = 3 - 1
[6,1,1]
=> 011 => 100 => 1 = 2 - 1
[5,3]
=> 11 => 00 => 0 = 1 - 1
[5,2,1]
=> 101 => 010 => 1 = 2 - 1
[5,1,1,1]
=> 1111 => 0000 => 0 = 1 - 1
[4,4]
=> 00 => 11 => 2 = 3 - 1
[4,3,1]
=> 011 => 100 => 1 = 2 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000093
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2]
=> 0 => [2] => ([],2)
=> 2
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4]
=> 0 => [2] => ([],2)
=> 2
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2]
=> 00 => [3] => ([],3)
=> 3
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1]
=> 1111 => [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)
=> 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1]
=> 11111 => [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)
=> 1
[6]
=> 0 => [2] => ([],2)
=> 2
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,2]
=> 00 => [3] => ([],3)
=> 3
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,1,1]
=> 1111 => [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)
=> 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 4
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1]
=> 01111 => [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)
=> 2
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 3
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,1,1,1]
=> 11111 => [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)
=> 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,2,1,1,1]
=> 00111 => [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)
=> 3
[8]
=> 0 => [2] => ([],2)
=> 2
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[6,2]
=> 00 => [3] => ([],3)
=> 3
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,1,1,1]
=> 1111 => [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)
=> 1
[4,4]
=> 00 => [3] => ([],3)
=> 3
[4,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(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: St000786
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2]
=> 0 => [2] => ([],2)
=> 2
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4]
=> 0 => [2] => ([],2)
=> 2
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2]
=> 00 => [3] => ([],3)
=> 3
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1]
=> 1111 => [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)
=> 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1]
=> 11111 => [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)
=> 1
[6]
=> 0 => [2] => ([],2)
=> 2
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,2]
=> 00 => [3] => ([],3)
=> 3
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,1,1]
=> 1111 => [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)
=> 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 4
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1]
=> 01111 => [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)
=> 2
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 3
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,1,1,1]
=> 11111 => [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)
=> 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,2,1,1,1]
=> 00111 => [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)
=> 3
[8]
=> 0 => [2] => ([],2)
=> 2
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[6,2]
=> 00 => [3] => ([],3)
=> 3
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,1,1,1]
=> 1111 => [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)
=> 1
[4,4]
=> 00 => [3] => ([],3)
=> 3
[4,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(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
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2]
=> 0 => [2] => ([],2)
=> 2
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4]
=> 0 => [2] => ([],2)
=> 2
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2]
=> 00 => [3] => ([],3)
=> 3
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1]
=> 1111 => [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)
=> 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1]
=> 11111 => [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)
=> 1
[6]
=> 0 => [2] => ([],2)
=> 2
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,2]
=> 00 => [3] => ([],3)
=> 3
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,1,1]
=> 1111 => [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)
=> 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 4
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1]
=> 01111 => [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)
=> 2
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 3
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,1,1,1]
=> 11111 => [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)
=> 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,2,1,1,1]
=> 00111 => [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)
=> 3
[8]
=> 0 => [2] => ([],2)
=> 2
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[6,2]
=> 00 => [3] => ([],3)
=> 3
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,1,1,1]
=> 1111 => [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)
=> 1
[4,4]
=> 00 => [3] => ([],3)
=> 3
[4,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(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
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2]
=> 0 => [2] => ([],2)
=> 2
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4]
=> 0 => [2] => ([],2)
=> 2
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,2]
=> 00 => [3] => ([],3)
=> 3
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1]
=> 1111 => [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)
=> 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1]
=> 11111 => [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)
=> 1
[6]
=> 0 => [2] => ([],2)
=> 2
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,2]
=> 00 => [3] => ([],3)
=> 3
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,1,1]
=> 1111 => [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)
=> 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 4
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1]
=> 01111 => [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)
=> 2
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 3
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,1,1,1]
=> 11111 => [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)
=> 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,2,1,1,1]
=> 00111 => [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)
=> 3
[8]
=> 0 => [2] => ([],2)
=> 2
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[6,2]
=> 00 => [3] => ([],3)
=> 3
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,1,1,1]
=> 1111 => [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)
=> 1
[4,4]
=> 00 => [3] => ([],3)
=> 3
[4,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(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: St000148
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[2]
=> [1,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> []
=> 0 = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0 = 1 - 1
[2,1]
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> []
=> 0 = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0 = 1 - 1
[2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> []
=> 0 = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0 = 1 - 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1 = 2 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2 = 3 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 0 = 1 - 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [2]
=> 0 = 1 - 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0 = 1 - 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1 = 2 - 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1 = 2 - 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2 = 3 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 0 = 1 - 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [2]
=> 0 = 1 - 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 1 = 2 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0 = 1 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 2 = 3 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1 = 2 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 0 = 1 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1 = 2 - 1
Description
The number of odd parts of a partition.
Matching statistic: St000992
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> []
=> 0 = 1 - 1
[2]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> 0 = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> 0 = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> 0 = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 2 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,3]
=> [2,2,2]
=> [2,2]
=> 0 = 1 - 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,2,2]
=> [3,3]
=> [3]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 1 = 2 - 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 2 - 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 0 = 1 - 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,2,2,1]
=> [4,3]
=> [3]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 2 = 3 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1 = 2 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 2 = 3 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 2 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 0 = 1 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 2 = 3 - 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[9,6]
=> [2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 2 - 1
[10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> ? = 3 - 1
[9,7]
=> [2,2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 - 1
[9,6,1]
=> [3,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 2 - 1
[9,5,2]
=> [3,3,2,2,2,1,1,1,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 2 - 1
[8,4,4]
=> [3,3,3,3,1,1,1,1]
=> [3,3,3,1,1,1,1]
=> ? = 4 - 1
[15,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2 - 1
[14,3]
=> [2,2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 2 - 1
[13,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1,1,1]
=> ? = 2 - 1
[13,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[12,5]
=> [2,2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1,1,1]
=> ? = 2 - 1
[12,3,2]
=> [3,3,2,1,1,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 1
[11,6]
=> [2,2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1,1]
=> ? = 2 - 1
[11,4,2]
=> [3,3,2,2,1,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> ? = 3 - 1
[11,2,2,2]
=> [4,4,1,1,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 4 - 1
[10,7]
=> [2,2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,2,1,1,1]
=> ? = 2 - 1
[10,6,1]
=> [3,2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> ? = 3 - 1
[10,5,2]
=> [3,3,2,2,2,1,1,1,1,1]
=> [3,2,2,2,1,1,1,1,1]
=> ? = 3 - 1
[10,4,3]
=> [3,3,3,2,1,1,1,1,1,1]
=> [3,3,2,1,1,1,1,1,1]
=> ? = 3 - 1
[10,3,2,2]
=> [4,4,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> ? = 4 - 1
[9,7,1]
=> [3,2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 - 1
[9,6,1,1]
=> [4,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 2 - 1
[9,5,3]
=> [3,3,3,2,2,1,1,1,1]
=> [3,3,2,2,1,1,1,1]
=> ? = 1 - 1
[9,5,2,1]
=> [4,3,2,2,2,1,1,1,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 2 - 1
[9,4,4]
=> [3,3,3,3,1,1,1,1,1]
=> [3,3,3,1,1,1,1,1]
=> ? = 3 - 1
[9,4,2,2]
=> [4,4,2,2,1,1,1,1,1]
=> [4,2,2,1,1,1,1,1]
=> ? = 4 - 1
[9,3,3,2]
=> [4,4,3,1,1,1,1,1,1]
=> [4,3,1,1,1,1,1,1]
=> ? = 2 - 1
[8,7,2]
=> [3,3,2,2,2,2,2,1]
=> [3,2,2,2,2,2,1]
=> ? = 3 - 1
[8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [3,3,2,2,2,1,1]
=> ? = 3 - 1
[8,5,4]
=> [3,3,3,3,2,1,1,1]
=> [3,3,3,2,1,1,1]
=> ? = 3 - 1
[8,5,2,2]
=> [4,4,2,2,2,1,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 4 - 1
[8,4,4,1]
=> [4,3,3,3,1,1,1,1]
=> [3,3,3,1,1,1,1]
=> ? = 4 - 1
[8,4,3,2]
=> [4,4,3,2,1,1,1,1]
=> [4,3,2,1,1,1,1]
=> ? = 4 - 1
[8,3,3,3]
=> [4,4,4,1,1,1,1,1]
=> [4,4,1,1,1,1,1]
=> ? = 2 - 1
[7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [4,3,2,2,1,1]
=> ? = 2 - 1
[7,4,4,2]
=> [4,4,3,3,1,1,1]
=> [4,3,3,1,1,1]
=> ? = 4 - 1
[7,4,3,3]
=> [4,4,4,2,1,1,1]
=> [4,4,2,1,1,1]
=> ? = 2 - 1
[6,6,3,2]
=> [4,4,3,2,2,2]
=> [4,3,2,2,2]
=> ? = 4 - 1
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000149
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
St000149: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0 = 1 - 1
[2]
=> [2]
=> 1 = 2 - 1
[1,1]
=> [1,1]
=> 0 = 1 - 1
[3]
=> [2,1]
=> 0 = 1 - 1
[2,1]
=> [3]
=> 1 = 2 - 1
[1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[4]
=> [2,2]
=> 1 = 2 - 1
[3,1]
=> [2,1,1]
=> 0 = 1 - 1
[2,2]
=> [4]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[5]
=> [2,2,1]
=> 0 = 1 - 1
[4,1]
=> [3,2]
=> 1 = 2 - 1
[3,2]
=> [4,1]
=> 1 = 2 - 1
[3,1,1]
=> [2,1,1,1]
=> 0 = 1 - 1
[2,2,1]
=> [5]
=> 2 = 3 - 1
[2,1,1,1]
=> [3,1,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[6]
=> [2,2,2]
=> 1 = 2 - 1
[5,1]
=> [2,2,1,1]
=> 0 = 1 - 1
[4,2]
=> [4,2]
=> 2 = 3 - 1
[4,1,1]
=> [4,1,1]
=> 1 = 2 - 1
[3,3]
=> [3,2,1]
=> 0 = 1 - 1
[3,2,1]
=> [3,3]
=> 1 = 2 - 1
[3,1,1,1]
=> [2,1,1,1,1]
=> 0 = 1 - 1
[2,2,2]
=> [6]
=> 3 = 4 - 1
[2,2,1,1]
=> [5,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [3,1,1,1]
=> 1 = 2 - 1
[7]
=> [2,2,2,1]
=> 0 = 1 - 1
[6,1]
=> [3,2,2]
=> 1 = 2 - 1
[5,2]
=> [4,2,1]
=> 1 = 2 - 1
[5,1,1]
=> [2,2,1,1,1]
=> 0 = 1 - 1
[4,3]
=> [4,3]
=> 1 = 2 - 1
[4,2,1]
=> [5,2]
=> 2 = 3 - 1
[4,1,1,1]
=> [4,1,1,1]
=> 1 = 2 - 1
[3,3,1]
=> [3,2,1,1]
=> 0 = 1 - 1
[3,2,2]
=> [6,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [3,3,1]
=> 1 = 2 - 1
[3,1,1,1,1]
=> [2,1,1,1,1,1]
=> 0 = 1 - 1
[2,2,2,1]
=> [7]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [5,1,1]
=> 2 = 3 - 1
[8]
=> [2,2,2,2]
=> 1 = 2 - 1
[7,1]
=> [2,2,2,1,1]
=> 0 = 1 - 1
[6,2]
=> [4,2,2]
=> 2 = 3 - 1
[6,1,1]
=> [4,2,1,1]
=> 1 = 2 - 1
[5,3]
=> [3,2,2,1]
=> 0 = 1 - 1
[5,2,1]
=> [3,3,2]
=> 1 = 2 - 1
[5,1,1,1]
=> [2,2,1,1,1,1]
=> 0 = 1 - 1
[4,4]
=> [4,4]
=> 2 = 3 - 1
[4,3,1]
=> [4,3,1]
=> 1 = 2 - 1
[3,2,2,2,1,1]
=> [7,3,1]
=> ? = 4 - 1
[10,1,1]
=> [4,2,2,2,1,1]
=> ? = 2 - 1
[7,3,1,1]
=> [3,2,2,2,1,1,1]
=> ? = 1 - 1
[7,2,1,1,1]
=> [4,4,1,1,1,1]
=> ? = 2 - 1
[5,5,2]
=> [6,3,2,1]
=> ? = 2 - 1
[5,5,1,1]
=> [4,3,2,1,1,1]
=> ? = 1 - 1
[4,4,1,1,1,1]
=> [7,2,1,1,1]
=> ? = 3 - 1
[11,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 1 - 1
[10,3]
=> [4,3,2,2,2]
=> ? = 2 - 1
[10,2,1]
=> [5,2,2,2,2]
=> ? = 3 - 1
[10,1,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 2 - 1
[9,2,1,1]
=> [4,4,2,1,1,1]
=> ? = 2 - 1
[7,5,1]
=> [4,3,2,2,1,1]
=> ? = 1 - 1
[7,3,1,1,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 1 - 1
[7,2,2,2]
=> [8,2,2,1]
=> ? = 4 - 1
[7,2,2,1,1]
=> [6,4,1,1,1]
=> ? = 3 - 1
[6,5,1,1]
=> [6,3,2,1,1]
=> ? = 2 - 1
[6,4,1,1,1]
=> [7,2,2,1,1]
=> ? = 3 - 1
[6,2,2,1,1,1]
=> [8,2,1,1,1]
=> ? = 4 - 1
[5,5,1,1,1]
=> [4,3,2,1,1,1,1]
=> ? = 1 - 1
[5,2,2,2,2]
=> [10,2,1]
=> ? = 5 - 1
[4,2,2,2,2,1]
=> [11,2]
=> ? = 6 - 1
[13,1]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 - 1
[12,2]
=> [4,2,2,2,2,2]
=> ? = 3 - 1
[12,1,1]
=> [4,2,2,2,2,1,1]
=> ? = 2 - 1
[11,3]
=> [3,2,2,2,2,2,1]
=> ? = 1 - 1
[11,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> ? = 1 - 1
[10,4]
=> [4,4,2,2,2]
=> ? = 3 - 1
[10,3,1]
=> [4,4,2,2,1,1]
=> ? = 2 - 1
[10,2,1,1]
=> [6,2,2,2,1,1]
=> ? = 3 - 1
[10,1,1,1,1]
=> [4,2,2,2,1,1,1,1]
=> ? = 2 - 1
[9,4,1]
=> [4,3,3,2,2]
=> ? = 2 - 1
[9,2,1,1,1]
=> [4,4,2,1,1,1,1]
=> ? = 2 - 1
[8,4,1,1]
=> [6,4,2,1,1]
=> ? = 3 - 1
[8,3,2,1]
=> [5,5,2,2]
=> ? = 3 - 1
[8,3,1,1,1]
=> [5,2,2,2,1,1,1]
=> ? = 2 - 1
[8,2,2,2]
=> [8,2,2,2]
=> ? = 5 - 1
[8,2,2,1,1]
=> [8,2,2,1,1]
=> ? = 4 - 1
[8,2,1,1,1,1]
=> [6,2,2,1,1,1,1]
=> ? = 3 - 1
[7,5,2]
=> [6,3,2,2,1]
=> ? = 2 - 1
[7,5,1,1]
=> [4,3,2,2,1,1,1]
=> ? = 1 - 1
[7,4,2,1]
=> [6,3,3,2]
=> ? = 3 - 1
[7,3,3,1]
=> [3,3,2,2,2,1,1]
=> ? = 1 - 1
[7,3,2,1,1]
=> [4,4,3,1,1,1]
=> ? = 2 - 1
[7,2,2,2,1]
=> [7,3,2,2]
=> ? = 4 - 1
[7,2,2,1,1,1]
=> [6,4,1,1,1,1]
=> ? = 3 - 1
[6,6,1,1]
=> [6,4,3,1]
=> ? = 3 - 1
[6,5,1,1,1]
=> [6,3,2,1,1,1]
=> ? = 2 - 1
[6,4,2,1,1]
=> [8,4,1,1]
=> ? = 4 - 1
[6,4,1,1,1,1]
=> [7,2,2,1,1,1]
=> ? = 3 - 1
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000150
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[1,1]
=> [1,1]
=> [2]
=> 0 = 1 - 1
[3]
=> [2,1]
=> [2,1]
=> 0 = 1 - 1
[2,1]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
[4]
=> [2,2]
=> [2,2]
=> 1 = 2 - 1
[3,1]
=> [2,1,1]
=> [3,1]
=> 0 = 1 - 1
[2,2]
=> [4]
=> [1,1,1,1]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[5]
=> [2,2,1]
=> [3,2]
=> 0 = 1 - 1
[4,1]
=> [3,2]
=> [2,2,1]
=> 1 = 2 - 1
[3,2]
=> [4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[3,1,1]
=> [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[2,2,1]
=> [5]
=> [1,1,1,1,1]
=> 2 = 3 - 1
[2,1,1,1]
=> [3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
[6]
=> [2,2,2]
=> [3,3]
=> 1 = 2 - 1
[5,1]
=> [2,2,1,1]
=> [4,2]
=> 0 = 1 - 1
[4,2]
=> [4,2]
=> [2,2,1,1]
=> 2 = 3 - 1
[4,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 1 = 2 - 1
[3,3]
=> [3,2,1]
=> [3,2,1]
=> 0 = 1 - 1
[3,2,1]
=> [3,3]
=> [2,2,2]
=> 1 = 2 - 1
[3,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 0 = 1 - 1
[2,2,2]
=> [6]
=> [1,1,1,1,1,1]
=> 3 = 4 - 1
[2,2,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [3,1,1,1]
=> [4,1,1]
=> 1 = 2 - 1
[7]
=> [2,2,2,1]
=> [4,3]
=> 0 = 1 - 1
[6,1]
=> [3,2,2]
=> [3,3,1]
=> 1 = 2 - 1
[5,2]
=> [4,2,1]
=> [3,2,1,1]
=> 1 = 2 - 1
[5,1,1]
=> [2,2,1,1,1]
=> [5,2]
=> 0 = 1 - 1
[4,3]
=> [4,3]
=> [2,2,2,1]
=> 1 = 2 - 1
[4,2,1]
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 3 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> 1 = 2 - 1
[3,3,1]
=> [3,2,1,1]
=> [4,2,1]
=> 0 = 1 - 1
[3,2,2]
=> [6,1]
=> [2,1,1,1,1,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [3,3,1]
=> [3,2,2]
=> 1 = 2 - 1
[3,1,1,1,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> 0 = 1 - 1
[2,2,2,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> 2 = 3 - 1
[8]
=> [2,2,2,2]
=> [4,4]
=> 1 = 2 - 1
[7,1]
=> [2,2,2,1,1]
=> [5,3]
=> 0 = 1 - 1
[6,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2 = 3 - 1
[6,1,1]
=> [4,2,1,1]
=> [4,2,1,1]
=> 1 = 2 - 1
[5,3]
=> [3,2,2,1]
=> [4,3,1]
=> 0 = 1 - 1
[5,2,1]
=> [3,3,2]
=> [3,3,2]
=> 1 = 2 - 1
[5,1,1,1]
=> [2,2,1,1,1,1]
=> [6,2]
=> 0 = 1 - 1
[4,4]
=> [4,4]
=> [2,2,2,2]
=> 2 = 3 - 1
[4,3,1]
=> [4,3,1]
=> [3,2,2,1]
=> 1 = 2 - 1
[5,3,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> ? = 1 - 1
[4,4,1,1,1]
=> [7,2,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 3 - 1
[8,2,2]
=> [6,2,2,2]
=> [4,4,1,1,1,1]
=> ? = 4 - 1
[6,4,1,1]
=> [6,4,1,1]
=> [4,2,2,2,1,1]
=> ? = 3 - 1
[5,5,2]
=> [6,3,2,1]
=> [4,3,2,1,1,1]
=> ? = 2 - 1
[5,5,1,1]
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> ? = 1 - 1
[5,2,2,2,1]
=> [7,3,2]
=> [3,3,2,1,1,1,1]
=> ? = 4 - 1
[4,3,3,2]
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> ? = 3 - 1
[10,2,1]
=> [5,2,2,2,2]
=> [5,5,1,1,1]
=> ? = 3 - 1
[8,3,2]
=> [6,3,2,2]
=> [4,4,2,1,1,1]
=> ? = 3 - 1
[8,3,1,1]
=> [5,2,2,2,1,1]
=> [6,4,1,1,1]
=> ? = 2 - 1
[8,2,2,1]
=> [7,2,2,2]
=> [4,4,1,1,1,1,1]
=> ? = 4 - 1
[7,4,2]
=> [6,4,2,1]
=> [4,3,2,2,1,1]
=> ? = 3 - 1
[7,2,2,2]
=> [8,2,2,1]
=> [4,3,1,1,1,1,1,1]
=> ? = 4 - 1
[5,4,2,2]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 4 - 1
[5,4,2,1,1]
=> [6,3,3,1]
=> [4,3,3,1,1,1]
=> ? = 3 - 1
[5,2,2,2,1,1]
=> [8,3,1,1]
=> [4,2,2,1,1,1,1,1]
=> ? = 4 - 1
[4,4,3,2]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 4 - 1
[4,4,3,1,1]
=> [7,4,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 3 - 1
[4,4,2,1,1,1]
=> [9,2,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> ? = 4 - 1
[4,3,3,3]
=> [5,5,2,1]
=> [4,3,2,2,2]
=> ? = 2 - 1
[3,3,3,2,2]
=> [7,3,2,1]
=> [4,3,2,1,1,1,1]
=> ? = 3 - 1
[3,3,2,2,2,1]
=> [7,3,3]
=> [3,3,3,1,1,1,1]
=> ? = 4 - 1
[12,1,1]
=> [4,2,2,2,2,1,1]
=> [7,5,1,1]
=> ? = 2 - 1
[10,4]
=> [4,4,2,2,2]
=> [5,5,2,2]
=> ? = 3 - 1
[10,2,1,1]
=> [6,2,2,2,1,1]
=> [6,4,1,1,1,1]
=> ? = 3 - 1
[10,1,1,1,1]
=> [4,2,2,2,1,1,1,1]
=> [8,4,1,1]
=> ? = 2 - 1
[9,5]
=> [4,3,2,2,2,1]
=> [6,5,2,1]
=> ? = 1 - 1
[9,4,1]
=> [4,3,3,2,2]
=> [5,5,3,1]
=> ? = 2 - 1
[9,2,2,1]
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> ? = 3 - 1
[9,2,1,1,1]
=> [4,4,2,1,1,1,1]
=> [7,3,2,2]
=> ? = 2 - 1
[8,4,2]
=> [6,4,2,2]
=> [4,4,2,2,1,1]
=> ? = 4 - 1
[8,3,2,1]
=> [5,5,2,2]
=> [4,4,2,2,2]
=> ? = 3 - 1
[8,3,1,1,1]
=> [5,2,2,2,1,1,1]
=> [7,4,1,1,1]
=> ? = 2 - 1
[8,2,2,2]
=> [8,2,2,2]
=> [4,4,1,1,1,1,1,1]
=> ? = 5 - 1
[8,2,1,1,1,1]
=> [6,2,2,1,1,1,1]
=> [7,3,1,1,1,1]
=> ? = 3 - 1
[7,5,1,1]
=> [4,3,2,2,1,1,1]
=> [7,4,2,1]
=> ? = 1 - 1
[7,4,2,1]
=> [6,3,3,2]
=> [4,4,3,1,1,1]
=> ? = 3 - 1
[7,4,1,1,1]
=> [5,4,2,1,1,1]
=> [6,3,2,2,1]
=> ? = 2 - 1
[7,3,2,1,1]
=> [4,4,3,1,1,1]
=> [6,3,3,2]
=> ? = 2 - 1
[7,2,2,2,1]
=> [7,3,2,2]
=> [4,4,2,1,1,1,1]
=> ? = 4 - 1
[7,2,2,1,1,1]
=> [6,4,1,1,1,1]
=> [6,2,2,2,1,1]
=> ? = 3 - 1
[6,6,1,1]
=> [6,4,3,1]
=> [4,3,3,2,1,1]
=> ? = 3 - 1
[6,5,3]
=> [5,4,4,1]
=> [4,3,3,3,1]
=> ? = 2 - 1
[6,5,1,1,1]
=> [6,3,2,1,1,1]
=> [6,3,2,1,1,1]
=> ? = 2 - 1
[6,4,3,1]
=> [6,6,1,1]
=> [4,2,2,2,2,2]
=> ? = 3 - 1
[6,4,2,2]
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 5 - 1
[6,4,2,1,1]
=> [8,4,1,1]
=> [4,2,2,2,1,1,1,1]
=> ? = 4 - 1
[6,4,1,1,1,1]
=> [7,2,2,1,1,1]
=> [6,3,1,1,1,1,1]
=> ? = 3 - 1
[6,3,3,2]
=> [7,4,2,1]
=> [4,3,2,2,1,1,1]
=> ? = 3 - 1
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000142
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000142: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0 = 1 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 0 = 1 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 0 = 1 - 1
[4]
=> 1 = 2 - 1
[3,1]
=> 0 = 1 - 1
[2,2]
=> 2 = 3 - 1
[2,1,1]
=> 1 = 2 - 1
[1,1,1,1]
=> 0 = 1 - 1
[5]
=> 0 = 1 - 1
[4,1]
=> 1 = 2 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 0 = 1 - 1
[2,2,1]
=> 2 = 3 - 1
[2,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> 0 = 1 - 1
[6]
=> 1 = 2 - 1
[5,1]
=> 0 = 1 - 1
[4,2]
=> 2 = 3 - 1
[4,1,1]
=> 1 = 2 - 1
[3,3]
=> 0 = 1 - 1
[3,2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> 0 = 1 - 1
[2,2,2]
=> 3 = 4 - 1
[2,2,1,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> 1 = 2 - 1
[7]
=> 0 = 1 - 1
[6,1]
=> 1 = 2 - 1
[5,2]
=> 1 = 2 - 1
[5,1,1]
=> 0 = 1 - 1
[4,3]
=> 1 = 2 - 1
[4,2,1]
=> 2 = 3 - 1
[4,1,1,1]
=> 1 = 2 - 1
[3,3,1]
=> 0 = 1 - 1
[3,2,2]
=> 2 = 3 - 1
[3,2,1,1]
=> 1 = 2 - 1
[3,1,1,1,1]
=> 0 = 1 - 1
[2,2,2,1]
=> 3 = 4 - 1
[2,2,1,1,1]
=> 2 = 3 - 1
[8]
=> 1 = 2 - 1
[7,1]
=> 0 = 1 - 1
[6,2]
=> 2 = 3 - 1
[6,1,1]
=> 1 = 2 - 1
[5,3]
=> 0 = 1 - 1
[5,2,1]
=> 1 = 2 - 1
[5,1,1,1]
=> 0 = 1 - 1
[4,4]
=> 2 = 3 - 1
[4,3,1]
=> 1 = 2 - 1
[9,2]
=> ? = 2 - 1
[8,2,1]
=> ? = 3 - 1
[8,1,1,1]
=> ? = 2 - 1
[7,3,1]
=> ? = 1 - 1
[7,2,1,1]
=> ? = 2 - 1
[7,1,1,1,1]
=> ? = 1 - 1
[6,3,2]
=> ? = 3 - 1
[6,3,1,1]
=> ? = 2 - 1
[6,2,2,1]
=> ? = 4 - 1
[6,2,1,1,1]
=> ? = 3 - 1
[4,2,2,1,1,1]
=> ? = 4 - 1
[3,2,2,2,1,1]
=> ? = 4 - 1
[9,2,1]
=> ? = 2 - 1
[9,1,1,1]
=> ? = 1 - 1
[8,4]
=> ? = 3 - 1
[8,3,1]
=> ? = 2 - 1
[8,2,1,1]
=> ? = 3 - 1
[8,1,1,1,1]
=> ? = 2 - 1
[7,3,2]
=> ? = 2 - 1
[7,3,1,1]
=> ? = 1 - 1
[7,2,2,1]
=> ? = 3 - 1
[7,2,1,1,1]
=> ? = 2 - 1
[6,5,1]
=> ? = 2 - 1
[6,4,1,1]
=> ? = 3 - 1
[6,3,2,1]
=> ? = 3 - 1
[6,3,1,1,1]
=> ? = 2 - 1
[6,2,1,1,1,1]
=> ? = 3 - 1
[5,5,1,1]
=> ? = 1 - 1
[4,4,1,1,1,1]
=> ? = 3 - 1
[4,3,2,1,1,1]
=> ? = 3 - 1
[4,2,2,2,2]
=> ? = 6 - 1
[4,2,2,2,1,1]
=> ? = 5 - 1
[11,2]
=> ? = 2 - 1
[11,1,1]
=> ? = 1 - 1
[10,2,1]
=> ? = 3 - 1
[10,1,1,1]
=> ? = 2 - 1
[9,4]
=> ? = 2 - 1
[9,3,1]
=> ? = 1 - 1
[9,2,1,1]
=> ? = 2 - 1
[9,1,1,1,1]
=> ? = 1 - 1
[8,4,1]
=> ? = 3 - 1
[8,3,1,1]
=> ? = 2 - 1
[8,2,2,1]
=> ? = 4 - 1
[8,2,1,1,1]
=> ? = 3 - 1
[7,4,1,1]
=> ? = 2 - 1
[7,3,3]
=> ? = 1 - 1
[7,3,2,1]
=> ? = 2 - 1
[7,3,1,1,1]
=> ? = 1 - 1
[7,2,2,2]
=> ? = 4 - 1
[7,2,2,1,1]
=> ? = 3 - 1
Description
The number of even parts of a partition.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000389The number of runs of ones of odd length in a binary word. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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