Identifier
Values
([],1) => [1] => [1,0] => [2,1] => 1
([],2) => [2] => [1,1,0,0] => [2,3,1] => 1
([(0,1)],2) => [1,1] => [1,0,1,0] => [3,1,2] => 2
([],3) => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 1
([(1,2)],3) => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 2
([(0,2),(1,2)],3) => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 3
([(0,1),(0,2),(1,2)],3) => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 2
([],4) => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
([(2,3)],4) => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
([(1,3),(2,3)],4) => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 3
([(0,3),(1,3),(2,3)],4) => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 3
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
([(0,3),(1,2),(2,3)],4) => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
([(1,2),(1,3),(2,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
([(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
([(0,2),(0,3),(1,2),(1,3)],4) => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
([],5) => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
([(3,4)],5) => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 2
([(2,4),(3,4)],5) => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 3
([(1,4),(2,4),(3,4)],5) => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 3
([(0,4),(1,4),(2,4),(3,4)],5) => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 3
([(1,4),(2,3)],5) => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 2
([(1,4),(2,3),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 4
([(0,1),(2,4),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 4
([(2,3),(2,4),(3,4)],5) => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 2
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 4
([(1,3),(1,4),(2,3),(2,4)],5) => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(0,1),(2,3),(2,4),(3,4)],5) => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 2
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Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is St000213The number of weak exceedances (also weak excedences) of a permutation., the number of deficiencies is St000703The number of deficiencies of a permutation..
Map
Laplacian multiplicities
Description
The composition of multiplicities of the Laplacian eigenvalues.
Let $\lambda_1 > \lambda_2 > \dots$ be the eigenvalues of the Laplacian matrix of a graph on $n$ vertices. Then this map returns the composition $a_1,\dots,a_k$ of $n$ where $a_i$ is the multiplicity of $\lambda_i$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
bounce path
Description
The bounce path determined by an integer composition.