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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St001130
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001130: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001130: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 3
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 0
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 3
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 0
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 0
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
Description
The number of two successive successions in a permutation.
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: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[2,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[3,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1 + 1
[4,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 2 + 1
[5,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1 + 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 2 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 3 + 1
[6,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1 + 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 0 + 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 2 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 3 + 1
[7,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1 + 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 2 + 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 0 + 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 2 + 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 0 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 3 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 1 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 2 + 1
[8,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1 + 1
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 2 + 1
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 0 + 1
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 2 + 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 + 1
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 0 + 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 3 + 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 1 + 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0 + 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 1 + 1
[9,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[10,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[11,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[12,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[11,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[13,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[12,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[12,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[14,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[13,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 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: St001964
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 25%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 25%
Values
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[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
[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)
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[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)
=> 0
[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
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0
[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)
=> 0
[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)
=> 0
[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)
=> 0
[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)
=> ? = 1
[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)
=> ? = 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [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
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[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)
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [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)
=> ? = 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 0
[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
[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)
=> ? = 2
[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)
=> ? = 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(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
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[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)
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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)
=> ? = 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[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)
=> 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[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
[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)
=> ? = 1
[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)
=> ? = 3
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,15),(4,6),(4,15),(5,9),(6,11),(7,5),(7,13),(9,10),(10,8),(11,2),(11,14),(12,8),(13,9),(13,14),(14,10),(14,12),(15,11),(15,13)],16)
=> ? = 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ([(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
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 2
[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)
=> 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[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)
=> 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[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)
=> 0
[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)
=> ? = 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ([(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)
=> ? = 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 1
[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
[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)
=> ? = 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,16),(4,8),(4,16),(5,13),(6,14),(7,5),(7,18),(8,6),(8,19),(10,11),(11,9),(12,9),(13,10),(14,2),(14,17),(15,10),(15,17),(16,18),(16,19),(17,11),(17,12),(18,13),(18,15),(19,14),(19,15)],20)
=> ? = 0
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,11),(3,12),(4,5),(4,15),(5,9),(6,3),(6,14),(7,4),(7,14),(8,1),(9,10),(10,8),(11,8),(12,2),(12,13),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 0
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,15),(4,6),(4,15),(5,9),(6,11),(7,5),(7,13),(9,10),(10,8),(11,2),(11,14),(12,8),(13,9),(13,14),(14,10),(14,12),(15,11),(15,13)],16)
=> ? = 0
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 0
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,3),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,1),(11,7),(12,2),(12,9)],13)
=> ? = 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(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)
=> ? = 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 0
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 0
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 0
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 2
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1
[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)
=> 0
[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)
=> 0
[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)
=> 0
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ([(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)
=> ? = 3
[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)
=> 0
[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)
=> 0
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001491
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Values
[1,1]
=> [1]
=> 10 => 00 => ? = 0 + 1
[2,1]
=> [1]
=> 10 => 00 => ? = 0 + 1
[1,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[3,1]
=> [1]
=> 10 => 00 => ? = 0 + 1
[2,2]
=> [2]
=> 100 => 000 => ? = 0 + 1
[2,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[4,1]
=> [1]
=> 10 => 00 => ? = 0 + 1
[3,2]
=> [2]
=> 100 => 000 => ? = 0 + 1
[3,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[2,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 2 + 1
[5,1]
=> [1]
=> 10 => 00 => ? = 0 + 1
[4,2]
=> [2]
=> 100 => 000 => ? = 0 + 1
[4,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[3,3]
=> [3]
=> 1000 => 0000 => ? = 1 + 1
[3,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[2,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 0 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 2 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 3 + 1
[6,1]
=> [1]
=> 10 => 00 => ? = 0 + 1
[5,2]
=> [2]
=> 100 => 000 => ? = 0 + 1
[5,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[4,3]
=> [3]
=> 1000 => 0000 => ? = 1 + 1
[4,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[3,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 0 + 1
[3,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 0 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 2 + 1
[2,2,2,1]
=> [2,2,1]
=> 11010 => 01010 => ? = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 001110 => ? = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 3 + 1
[7,1]
=> [1]
=> 10 => 00 => ? = 0 + 1
[6,2]
=> [2]
=> 100 => 000 => ? = 0 + 1
[6,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[5,3]
=> [3]
=> 1000 => 0000 => ? = 1 + 1
[5,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[4,4]
=> [4]
=> 10000 => 00000 => ? = 2 + 1
[4,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 0 + 1
[4,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 0 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 2 + 1
[3,3,2]
=> [3,2]
=> 10100 => 00100 => ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> 100110 => 000110 => ? = 0 + 1
[3,2,2,1]
=> [2,2,1]
=> 11010 => 01010 => ? = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 001110 => ? = 1 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 3 + 1
[2,2,2,2]
=> [2,2,2]
=> 11100 => 01100 => ? = 1 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 010110 => ? = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0011110 => ? = 2 + 1
[8,1]
=> [1]
=> 10 => 00 => ? = 0 + 1
[7,2]
=> [2]
=> 100 => 000 => ? = 0 + 1
[7,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[6,3]
=> [3]
=> 1000 => 0000 => ? = 1 + 1
[6,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[5,4]
=> [4]
=> 10000 => 00000 => ? = 2 + 1
[5,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 0 + 1
[5,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 0 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 2 + 1
[4,4,1]
=> [4,1]
=> 100010 => 000010 => ? = 1 + 1
[4,3,2]
=> [3,2]
=> 10100 => 00100 => ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> 100110 => 000110 => ? = 0 + 1
[4,2,2,1]
=> [2,2,1]
=> 11010 => 01010 => ? = 0 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 001110 => ? = 1 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 3 + 1
[8,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[7,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[7,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[6,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[9,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[8,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[8,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[7,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[10,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[9,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[9,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[8,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[11,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[10,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[10,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[9,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[12,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[11,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[11,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[10,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[13,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[12,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[12,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[11,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[14,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[13,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[13,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[12,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000782
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[3,1]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 2 + 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 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 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 2 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 3 + 1
[6,1]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 0 + 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 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 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 2 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 3 + 1
[7,1]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 + 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 0 + 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 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 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 2 + 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 0 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 3 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 2 + 1
[8,1]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 2 + 1
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 0 + 1
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 2 + 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> ? = 1 + 1
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[11,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[12,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[12,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[13,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[14,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[14,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[13,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[15,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[15,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[14,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 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: St000181
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
[1,1]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,1]
=> [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 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,1,1]
=> [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)
=> ? = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[4,1,1]
=> [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
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[2,1,1,1,1]
=> [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)
=> ? = 2 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(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)
=> ? = 3 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ([(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
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 0 + 1
[5,1,1]
=> [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 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,1,1,1,1]
=> [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)
=> ? = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 1 + 1
[2,1,1,1,1,1]
=> [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)
=> ? = 3 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 0 + 1
[6,1,1]
=> [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 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 1
[5,1,1,1]
=> [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)
=> ? = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,1,1,1]
=> [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)
=> ? = 2 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 1 + 1
[3,1,1,1,1,1]
=> [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)
=> ? = 3 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 2 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ([(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 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,11),(3,12),(4,5),(4,15),(5,9),(6,3),(6,14),(7,4),(7,14),(8,1),(9,10),(10,8),(11,8),(12,2),(12,13),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 0 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,11),(3,12),(4,5),(4,15),(5,9),(6,3),(6,14),(7,4),(7,14),(8,1),(9,10),(10,8),(11,8),(12,2),(12,13),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ([(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)
=> ? = 1 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 0 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 0 + 1
[5,1,1,1,1]
=> [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)
=> ? = 2 + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(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)
=> ? = 3 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001490
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? = 2 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ? = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ? = 2 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ? = 3 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ? = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ? = 0 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ? = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ? = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ? = 3 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ? = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ? = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ? = 0 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ? = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ? = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ? = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ? = 2 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 1 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> ? = 2 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> ? = 1 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3,1],[]]
=> ? = 3 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ? = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,1],[1]]
=> ? = 2 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,4],[3]]
=> ? = 0 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3,1]]
=> ? = 0 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> ? = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1]]
=> ? = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> ? = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> ? = 1 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> ? = 2 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> ? = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ? = 0 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> ? = 0 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ? = 2 + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ? = 1 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> ? = 1 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3,1],[1]]
=> ? = 3 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ? = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St001890
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
[1,1]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,1]
=> [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 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,1,1]
=> [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)
=> ? = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[4,1,1]
=> [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
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[2,1,1,1,1]
=> [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)
=> ? = 2 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(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)
=> ? = 3 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ([(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
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 0 + 1
[5,1,1]
=> [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 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,1,1,1,1]
=> [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)
=> ? = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 1 + 1
[2,1,1,1,1,1]
=> [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)
=> ? = 3 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,5),(2,15),(3,12),(4,11),(5,4),(5,13),(6,2),(6,14),(7,3),(7,14),(8,1),(9,10),(10,8),(11,8),(12,9),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 0 + 1
[6,1,1]
=> [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 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0 + 1
[5,1,1,1]
=> [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)
=> ? = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,1,1,1]
=> [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)
=> ? = 2 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 1 + 1
[3,1,1,1,1,1]
=> [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)
=> ? = 3 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? = 2 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ([(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 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,11),(3,12),(4,5),(4,15),(5,9),(6,3),(6,14),(7,4),(7,14),(8,1),(9,10),(10,8),(11,8),(12,2),(12,13),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 0 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,11),(3,12),(4,5),(4,15),(5,9),(6,3),(6,14),(7,4),(7,14),(8,1),(9,10),(10,8),(11,8),(12,2),(12,13),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ([(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)
=> ? = 1 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 0 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 0 + 1
[5,1,1,1,1]
=> [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)
=> ? = 2 + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(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)
=> ? = 3 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 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.
Matching statistic: St000635
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000635: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000635: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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 + 1
[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 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 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 = 0 + 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 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [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
[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 = 0 + 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 = 0 + 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 = 0 + 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)
=> ? = 1 + 1
[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)
=> ? = 2 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [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 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1 + 1
[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 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [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)
=> ? = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 0 + 1
[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
[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)
=> ? = 2 + 1
[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)
=> ? = 3 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(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 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 0 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1 + 1
[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 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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)
=> ? = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 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 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2 + 1
[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 + 1
[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)
=> ? = 1 + 1
[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)
=> ? = 3 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,15),(4,6),(4,15),(5,9),(6,11),(7,5),(7,13),(9,10),(10,8),(11,2),(11,14),(12,8),(13,9),(13,14),(14,10),(14,12),(15,11),(15,13)],16)
=> ? = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ([(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 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 2 + 1
[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 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[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 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[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 = 0 + 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)
=> ? = 1 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ([(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)
=> ? = 3 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 1 + 1
[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 + 1
[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)
=> ? = 2 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,16),(4,8),(4,16),(5,13),(6,14),(7,5),(7,18),(8,6),(8,19),(10,11),(11,9),(12,9),(13,10),(14,2),(14,17),(15,10),(15,17),(16,18),(16,19),(17,11),(17,12),(18,13),(18,15),(19,14),(19,15)],20)
=> ? = 0 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ([(0,6),(0,7),(2,11),(3,12),(4,5),(4,15),(5,9),(6,3),(6,14),(7,4),(7,14),(8,1),(9,10),(10,8),(11,8),(12,2),(12,13),(13,10),(13,11),(14,12),(14,15),(15,9),(15,13)],16)
=> ? = 0 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ([(0,1),(1,3),(1,4),(2,12),(3,7),(3,15),(4,6),(4,15),(5,9),(6,11),(7,5),(7,13),(9,10),(10,8),(11,2),(11,14),(12,8),(13,9),(13,14),(14,10),(14,12),(15,11),(15,13)],16)
=> ? = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
=> ? = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,3),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,1),(11,7),(12,2),(12,9)],13)
=> ? = 1 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(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)
=> ? = 2 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 0 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 0 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 2 + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 + 1
[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 = 0 + 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 = 0 + 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 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ([(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)
=> ? = 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]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 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 = 0 + 1
Description
The number of strictly order preserving maps of a poset into itself.
A map $f$ is strictly order preserving if $a < b$ implies $f(a) < f(b)$.
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