Identifier
Values
[1] => [1,0] => [[]] => ([(0,1)],2) => 1
[2] => [1,0,1,0] => [[],[]] => ([(0,2),(1,2)],3) => 2
[1,1] => [1,1,0,0] => [[[]]] => ([(0,2),(1,2)],3) => 2
[3] => [1,0,1,0,1,0] => [[],[],[]] => ([(0,3),(1,3),(2,3)],4) => 3
[2,1] => [1,0,1,1,0,0] => [[],[[]]] => ([(0,3),(1,2),(2,3)],4) => 2
[1,1,1] => [1,1,0,1,0,0] => [[[],[]]] => ([(0,3),(1,3),(2,3)],4) => 3
[4] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[3,1] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[2,2] => [1,1,1,0,0,0] => [[[[]]]] => ([(0,3),(1,2),(2,3)],4) => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [[],[[],[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [[[],[],[]]] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[5] => [1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 4
[3,2] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3
[2,2,1] => [1,1,1,0,0,1,0,0] => [[[[]],[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 4
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [[[],[],[],[]]] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 5
[6] => [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
[5,1] => [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) => 5
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [[],[],[],[[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 4
[3,3] => [1,1,1,0,1,0,0,0] => [[[[],[]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 3
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [[],[[[]],[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [[],[],[[],[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 4
[2,2,2] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [[[[]],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 4
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [[],[[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 5
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[[],[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 6
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[],[]] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 7
[5,2] => [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) => 5
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [[],[],[[[]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 4
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 3
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [[],[[[]],[],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 4
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [[[[[]]],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[[[]],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 5
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[[],[],[],[],[],[]]] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 7
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [[],[],[[[],[]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 5
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 4
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [[],[[[],[]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 4
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [[[[],[]],[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 4
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [[],[[[[]]],[]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [[[[[]]],[],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 5
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [[],[[[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 5
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [[[[],[],[]],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 4
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [[],[[[],[[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 3
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[[[],[[]]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 4
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [[],[[[[],[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[[],[]]],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 5
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [[[[],[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 5
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [[[[],[],[[]]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => 4
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[],[[[[[]]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 4
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [[[[[[]]]],[]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [[[[],[[],[]]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => 4
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [[[[[],[],[]]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 5
[5,3,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [[],[],[[[],[[]]],[]]] => ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9) => 5
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [[[[],[[[]]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[4,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [[],[],[[[[],[]]],[]]] => ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9) => 5
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [[[[[[]],[]]]]] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 4
[5,3,2,2] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [[],[],[[[],[[],[]]]]] => ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9) => 5
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [[[[[[],[]]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => 4
[4,3,2,2,1] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [[],[[[],[[],[]]],[]]] => ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9) => 5
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[[[[[]]]]]]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 4
[4,4,3,2,1] => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0] => [[[[],[[[]],[]]],[]]] => ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9) => 5
[5,5,4,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0] => [[[[],[[[],[]]]],[]]] => ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9) => 5
[4,4,4,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [[[[[[],[]],[]]],[]]] => ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9) => 5
[5,5,4,2] => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [[[[],[[[],[]],[]]]]] => ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9) => 5
[] => [] => [] => ([],1) => 1
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Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges.