Identifier
Values
[1] => [[1],[]] => ([],1) => 2
[2] => [[2],[]] => ([(0,1)],2) => 3
[1,1] => [[1,1],[]] => ([(0,1)],2) => 3
[3] => [[3],[]] => ([(0,2),(2,1)],3) => 4
[2,1] => [[2,1],[]] => ([(0,1),(0,2)],3) => 5
[1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 4
[4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 5
[3,1] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 7
[2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 6
[2,1,1] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 7
[1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 5
[5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 6
[4,1] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 9
[3,2] => [[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 9
[3,1,1] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 10
[2,2,1] => [[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 9
[2,1,1,1] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 9
[1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 6
[6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 7
[5,1] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 11
[4,2] => [[4,2],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => 12
[4,1,1] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 13
[3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 10
[3,2,1] => [[3,2,1],[]] => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => 14
[3,1,1,1] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 13
[2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 10
[2,2,1,1] => [[2,2,1,1],[]] => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => 12
[2,1,1,1,1] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 11
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 7
[7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 8
[6,1] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => 13
[5,2] => [[5,2],[]] => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7) => 15
[5,1,1] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 16
[4,3] => [[4,3],[]] => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7) => 14
[4,2,1] => [[4,2,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7) => 19
[4,1,1,1] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => 17
[3,3,1] => [[3,3,1],[]] => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7) => 16
[3,2,2] => [[3,2,2],[]] => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7) => 16
[3,2,1,1] => [[3,2,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7) => 19
[3,1,1,1,1] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 16
[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) => 14
[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) => 15
[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) => 13
[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) => 8
[8] => [[8],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 9
[7,1] => [[7,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => 15
[6,2] => [[6,2],[]] => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8) => 18
[6,1,1] => [[6,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) => 19
[5,3] => [[5,3],[]] => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8) => 18
[5,2,1] => [[5,2,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8) => 24
[5,1,1,1] => [[5,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) => 21
[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) => 15
[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) => 23
[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) => 22
[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) => 26
[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) => 21
[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) => 19
[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) => 22
[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) => 23
[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) => 24
[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) => 19
[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) => 15
[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) => 18
[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) => 18
[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) => 15
[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) => 9
[9] => [[9],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 10
[8,1] => [[8,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => 17
[7,2] => [[7,2],[]] => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9) => 21
[7,1,1] => [[7,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => 22
[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) => 22
[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) => 29
[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) => 25
[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) => 20
[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) => 30
[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) => 28
[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) => 33
[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) => 26
[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) => 25
[4,3,2] => [[4,3,2],[]] => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9) => 28
[4,3,1,1] => [[4,3,1,1],[]] => ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9) => 32
[4,2,2,1] => [[4,2,2,1],[]] => ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9) => 32
[4,2,1,1,1] => [[4,2,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8)],9) => 33
[4,1,1,1,1,1] => [[4,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9) => 25
[3,3,3] => [[3,3,3],[]] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 20
[3,3,2,1] => [[3,3,2,1],[]] => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9) => 28
[3,3,1,1,1] => [[3,3,1,1,1],[]] => ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9) => 28
[3,2,2,2] => [[3,2,2,2],[]] => ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9) => 25
[3,2,2,1,1] => [[3,2,2,1,1],[]] => ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9) => 30
[3,2,1,1,1,1] => [[3,2,1,1,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(6,8),(7,1),(7,8)],9) => 29
[3,1,1,1,1,1,1] => [[3,1,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => 22
[2,2,2,2,1] => [[2,2,2,2,1],[]] => ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9) => 20
[2,2,2,1,1,1] => [[2,2,2,1,1,1],[]] => ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9) => 22
[2,2,1,1,1,1,1] => [[2,2,1,1,1,1,1],[]] => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9) => 21
[2,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => 17
[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 10
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Description
The number of antichains in a poset.
An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable.
An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Map
to skew partition
Description
The partition regarded as a skew partition.
Map
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.