Identifier
Values
[1] => [1,0] => [[]] => ([(0,1)],2) => 1
[1,1] => [1,0,1,0] => [[],[]] => ([(0,2),(1,2)],3) => 2
[2] => [1,1,0,0] => [[[]]] => ([(0,2),(2,1)],3) => 1
[1,1,1] => [1,0,1,0,1,0] => [[],[],[]] => ([(0,3),(1,3),(2,3)],4) => 3
[1,2] => [1,0,1,1,0,0] => [[],[[]]] => ([(0,3),(1,2),(2,3)],4) => 1
[2,1] => [1,1,0,0,1,0] => [[[]],[]] => ([(0,3),(1,2),(2,3)],4) => 1
[3] => [1,1,1,0,0,0] => [[[[]]]] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[1,1,2] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,2,1] => [1,0,1,1,0,0,1,0] => [[],[[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,3] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1
[2,1,1] => [1,1,0,0,1,0,1,0] => [[[]],[],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[2,2] => [1,1,0,0,1,1,0,0] => [[[]],[[]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[3,1] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1
[4] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 1
[5] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[[]],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 4
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[[]],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[[[]]],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [[],[[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [[],[[]],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[],[[]],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[],[[[]]],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[],[[[[]]]],[]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[[]],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[[]],[],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[[]],[],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[[]],[[]],[],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[[[]]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 3
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[[[]]]],[],[]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => 2
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Description
The distinguishing number of a poset.
This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring.
See also St000469The distinguishing number of a graph., which is the same concept for graphs.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to poset
Description
Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.
Map
to ordered tree
Description
Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path $D$ of semilength $n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths $D_1,\dots,D_k$ of respective semilengths $n_1,\dots,n_k$ (so one has $n = n_1 + \dots n_k$) each of which has no returns.
Denote by $\tilde D_i$ the path of semilength $n_i-1$ obtained from $D_i$ by removing the initial up- and the final down-step.
This map then sends $D$ to the tree $T$ having a root note with ordered children $T_1,\dots,T_k$ which are again ordered trees computed from $D_1,\dots,D_k$ respectively.
The unique path of semilength $1$ is sent to the tree consisting of a single node.