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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000618
St000618: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 0
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 0
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 3
[4,1,1]
=> 2
[3,3]
=> 3
[3,2,1]
=> 0
[3,1,1,1]
=> 2
[2,2,2]
=> 3
[2,2,1,1]
=> 3
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1
[7]
=> 1
[6,1]
=> 0
[5,2]
=> 2
[5,1,1]
=> 3
[4,3]
=> 0
[4,2,1]
=> 1
[4,1,1,1]
=> 0
[3,3,1]
=> 3
[3,2,2]
=> 3
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 0
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 1
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 4
[6,1,1]
=> 3
[5,3]
=> 4
[5,2,1]
=> 0
Description
The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St001722
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 14%●distinct values known / distinct values provided: 5%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 14%●distinct values known / distinct values provided: 5%
Values
[1]
=> []
=> []
=> => ? = 1
[2]
=> []
=> []
=> => ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[3]
=> []
=> []
=> => ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[4]
=> []
=> []
=> => ? ∊ {1,2}
[3,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {1,2}
[5]
=> []
=> []
=> => ? ∊ {0,0,2}
[4,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {0,0,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,0,2}
[6]
=> []
=> []
=> => ? ∊ {0,2,2,3,3,3,3}
[5,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? ∊ {0,2,2,3,3,3,3}
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {0,2,2,3,3,3,3}
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? ∊ {0,2,2,3,3,3,3}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? ∊ {0,2,2,3,3,3,3}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,2,2,3,3,3,3}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {0,2,2,3,3,3,3}
[7]
=> []
=> []
=> => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[6,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[8]
=> []
=> []
=> => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[9]
=> []
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[8,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[9,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[10,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[11,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000782
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 5% ●values known / values provided: 11%●distinct values known / distinct values provided: 5%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 5% ●values known / values provided: 11%●distinct values known / distinct values provided: 5%
Values
[1]
=> []
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> []
=> ? ∊ {1,1}
[1,1]
=> [1]
=> [1,0]
=> [(1,2)]
=> ? ∊ {1,1}
[3]
=> []
=> []
=> []
=> ? ∊ {0,1,1}
[2,1]
=> [1]
=> [1,0]
=> [(1,2)]
=> ? ∊ {0,1,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? ∊ {0,1,1}
[4]
=> []
=> []
=> []
=> ? ∊ {1,1,1,2}
[3,1]
=> [1]
=> [1,0]
=> [(1,2)]
=> ? ∊ {1,1,1,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? ∊ {1,1,1,2}
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? ∊ {1,1,1,2}
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[5]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1,2}
[4,1]
=> [1]
=> [1,0]
=> [(1,2)]
=> ? ∊ {0,0,1,1,2}
[3,2]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? ∊ {0,0,1,1,2}
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? ∊ {0,0,1,1,2}
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? ∊ {0,0,1,1,2}
[6]
=> []
=> []
=> []
=> ? ∊ {0,2,2,3,3,3,3}
[5,1]
=> [1]
=> [1,0]
=> [(1,2)]
=> ? ∊ {0,2,2,3,3,3,3}
[4,2]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? ∊ {0,2,2,3,3,3,3}
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? ∊ {0,2,2,3,3,3,3}
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? ∊ {0,2,2,3,3,3,3}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? ∊ {0,2,2,3,3,3,3}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> ? ∊ {0,2,2,3,3,3,3}
[7]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[6,1]
=> [1]
=> [1,0]
=> [(1,2)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[5,2]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3}
[8]
=> []
=> []
=> []
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,1]
=> [1]
=> [1,0]
=> [(1,2)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,2]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? ∊ {0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[7,3]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[8,3]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[8,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[9,3]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001510
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001510: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Mp00185: Skew partitions —cell poset⟶ Posets
St001510: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 21%
Values
[1]
=> [[1],[]]
=> ([],1)
=> 1
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 0
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 0
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 0
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 3
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 0
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 3
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,3,3,3,3}
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[9]
=> [[9],[]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[8,1]
=> [[8,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,2]
=> [[7,2],[]]
=> ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,1,1]
=> [[7,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,3]
=> [[6,3],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,2,1]
=> [[6,2,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(6,8),(7,1),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,1,1,1]
=> [[6,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,4]
=> [[5,4],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,3,1]
=> [[5,3,1],[]]
=> ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,2,2]
=> [[5,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,2,1,1]
=> [[5,2,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,4,1]
=> [[4,4,1],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
Description
The number of self-evacuating linear extensions of a finite poset.
Matching statistic: St001330
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 10%●distinct values known / distinct values provided: 5%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 10%●distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2} + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2} + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,2} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2} + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,2} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,0,1,2} + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,2,3,3,3,3} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,3,3,3,3} + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,2,3,3,3,3} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,2,3,3,3,3} + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {0,2,2,3,3,3,3} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,2,3,3,3,3} + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> 2 = 1 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 1 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3} + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6)],8)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => ([(0,4),(1,4),(2,6),(2,7),(3,6),(3,7),(4,5),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => ([(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(5,6),(5,7)],8)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,2,8,3,4,5,6,7] => ([(0,8),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,8,7,1,3,4,5,6] => ([(0,5),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,8,2,3,4,5,6,7] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)],9)
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,1,2,3,4,5,8,9,6] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [10,1,2,3,4,5,6,9,7,8] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [8,1,2,3,6,7,4,5] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8} + 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 1 + 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000181
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 9%●distinct values known / distinct values provided: 5%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 9%●distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {1,2}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,2}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {0,0,1,2}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,0,1,2}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {0,0,1,2}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {0,0,1,2}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,15),(4,9),(5,10),(6,3),(6,14),(7,2),(7,14),(8,1),(9,11),(10,12),(11,8),(12,8),(13,11),(13,12),(14,15),(14,16),(15,9),(15,13),(16,10),(16,13)],17)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ([(0,7),(0,8),(2,5),(2,17),(3,6),(3,16),(4,14),(5,13),(6,4),(6,20),(7,2),(7,18),(8,3),(8,18),(9,1),(10,12),(11,9),(12,9),(13,10),(14,11),(15,10),(15,19),(16,15),(16,20),(17,13),(17,15),(18,16),(18,17),(19,11),(19,12),(20,14),(20,19)],21)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,6),(5,8),(6,2),(6,11),(7,4),(7,5),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,1)],14)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,6),(2,8),(3,9),(4,10),(5,3),(5,7),(6,5),(6,10),(7,8),(7,9),(8,11),(9,11),(10,2),(10,7),(11,1)],12)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ([(0,7),(0,8),(1,6),(1,19),(2,4),(2,18),(3,14),(4,15),(5,3),(5,23),(6,5),(6,22),(7,1),(7,20),(8,2),(8,20),(10,11),(11,12),(12,9),(13,9),(14,13),(15,10),(16,12),(16,13),(17,11),(17,16),(18,15),(18,21),(19,21),(19,22),(20,18),(20,19),(21,10),(21,17),(22,17),(22,23),(23,14),(23,16)],24)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ([(0,8),(0,9),(2,7),(2,20),(3,6),(3,19),(4,15),(5,16),(6,4),(6,24),(7,5),(7,25),(8,3),(8,21),(9,2),(9,21),(10,1),(11,13),(12,14),(13,10),(14,10),(15,11),(16,12),(17,22),(17,23),(18,13),(18,14),(19,17),(19,24),(20,17),(20,25),(21,19),(21,20),(22,11),(22,18),(23,12),(23,18),(24,15),(24,22),(25,16),(25,23)],26)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,2),(2,3),(2,4),(3,8),(3,15),(4,7),(4,15),(5,12),(6,13),(7,5),(7,16),(8,6),(8,17),(9,1),(10,9),(11,9),(12,10),(13,11),(14,10),(14,11),(15,16),(15,17),(16,12),(16,14),(17,13),(17,14)],18)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St000772
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 1
[2]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 1
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1}
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,1}
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,2}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,2}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,1,1}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,1,1}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5,8] => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => ([(1,2),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6,8] => ([(1,4),(2,3),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
[4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1
[5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001490
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 5% ●values known / values provided: 9%●distinct values known / distinct values provided: 5%
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 5% ●values known / values provided: 9%●distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,2}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {1,2}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? ∊ {0,0,1,2}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {0,0,1,2}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {0,0,1,2}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> ? ∊ {0,0,1,2}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2],[]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5],[]]
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2],[1]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[5,5],[1]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1,1]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,4],[]]
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[4,3,2],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [[5,5,4],[1]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,2],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5,3],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [[6,6,6,6],[]]
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,5],[4]]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4,1]]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3,1]]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1
Description
The number of connected components of a skew partition.
Matching statistic: St001890
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 9%●distinct values known / distinct values provided: 5%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 9%●distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {1,2}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,2}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {0,0,1,2}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,0,1,2}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {0,0,1,2}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {0,0,1,2}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? ∊ {0,1,1,2,2,3,3,3,3}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,15),(4,9),(5,10),(6,3),(6,14),(7,2),(7,14),(8,1),(9,11),(10,12),(11,8),(12,8),(13,11),(13,12),(14,15),(14,16),(15,9),(15,13),(16,10),(16,13)],17)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
=> ? ∊ {0,0,0,0,0,1,2,2,3,3,3,3}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ([(0,7),(0,8),(2,5),(2,17),(3,6),(3,16),(4,14),(5,13),(6,4),(6,20),(7,2),(7,18),(8,3),(8,18),(9,1),(10,12),(11,9),(12,9),(13,10),(14,11),(15,10),(15,19),(16,15),(16,20),(17,13),(17,15),(18,16),(18,17),(19,11),(19,12),(20,14),(20,19)],21)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,7),(2,10),(3,9),(4,3),(4,8),(5,6),(5,8),(6,2),(6,11),(7,4),(7,5),(8,9),(8,11),(9,12),(10,13),(11,10),(11,12),(12,13),(13,1)],14)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,6),(2,8),(3,9),(4,10),(5,3),(5,7),(6,5),(6,10),(7,8),(7,9),(8,11),(9,11),(10,2),(10,7),(11,1)],12)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ([(0,7),(0,8),(1,6),(1,19),(2,4),(2,18),(3,14),(4,15),(5,3),(5,23),(6,5),(6,22),(7,1),(7,20),(8,2),(8,20),(10,11),(11,12),(12,9),(13,9),(14,13),(15,10),(16,12),(16,13),(17,11),(17,16),(18,15),(18,21),(19,21),(19,22),(20,18),(20,19),(21,10),(21,17),(22,17),(22,23),(23,14),(23,16)],24)
=> ? ∊ {0,0,1,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ([(0,8),(0,9),(2,7),(2,20),(3,6),(3,19),(4,15),(5,16),(6,4),(6,24),(7,5),(7,25),(8,3),(8,21),(9,2),(9,21),(10,1),(11,13),(12,14),(13,10),(14,10),(15,11),(16,12),(17,22),(17,23),(18,13),(18,14),(19,17),(19,24),(20,17),(20,25),(21,19),(21,20),(22,11),(22,18),(23,12),(23,18),(24,15),(24,22),(25,16),(25,23)],26)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,2),(2,3),(2,4),(3,8),(3,15),(4,7),(4,15),(5,12),(6,13),(7,5),(7,16),(8,6),(8,17),(9,1),(10,9),(11,9),(12,10),(13,11),(14,10),(14,11),(15,16),(15,17),(16,12),(16,14),(17,13),(17,14)],18)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,10),(3,9),(4,3),(4,8),(5,2),(5,8),(6,7),(7,4),(7,5),(8,9),(8,10),(9,11),(10,11),(11,1)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8}
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
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