Identifier
-
Mp00037:
Graphs
—to partition of connected components⟶
Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤ
Values
([],3) => [1,1,1] => [1,1] => 1
([],4) => [1,1,1,1] => [1,1,1] => 3
([(2,3)],4) => [2,1,1] => [1,1] => 1
([(0,3),(1,2)],4) => [2,2] => [2] => 0
([],5) => [1,1,1,1,1] => [1,1,1,1] => 6
([(3,4)],5) => [2,1,1,1] => [1,1,1] => 3
([(2,4),(3,4)],5) => [3,1,1] => [1,1] => 1
([(1,4),(2,3)],5) => [2,2,1] => [2,1] => 2
([(0,1),(2,4),(3,4)],5) => [3,2] => [2] => 0
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [2] => 0
([],6) => [1,1,1,1,1,1] => [1,1,1,1,1] => 10
([(4,5)],6) => [2,1,1,1,1] => [1,1,1,1] => 6
([(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => 3
([(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
([(2,5),(3,4)],6) => [2,2,1,1] => [2,1,1] => 5
([(2,5),(3,4),(4,5)],6) => [4,1,1] => [1,1] => 1
([(1,2),(3,5),(4,5)],6) => [3,2,1] => [2,1] => 2
([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => 3
([(0,1),(2,5),(3,5),(4,5)],6) => [4,2] => [2] => 0
([(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
([(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,1] => 1
([(0,5),(1,5),(2,4),(3,4)],6) => [3,3] => [3] => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [2,2] => 4
([(0,1),(2,5),(3,4),(4,5)],6) => [4,2] => [2] => 0
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [2,1] => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [2] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => [2] => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,3] => [3] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [2] => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [3] => 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [2] => 0
([],7) => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 15
([(5,6)],7) => [2,1,1,1,1,1] => [1,1,1,1,1] => 10
([(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => 6
([(3,6),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 3
([(2,6),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(3,6),(4,5)],7) => [2,2,1,1,1] => [2,1,1,1] => 9
([(3,6),(4,5),(5,6)],7) => [4,1,1,1] => [1,1,1] => 3
([(2,3),(4,6),(5,6)],7) => [3,2,1,1] => [2,1,1] => 5
([(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => 6
([(2,6),(3,6),(4,5),(5,6)],7) => [5,1,1] => [1,1] => 1
([(1,2),(3,6),(4,6),(5,6)],7) => [4,2,1] => [2,1] => 2
([(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1,1] => [1,1,1] => 3
([(1,6),(2,6),(3,5),(4,5)],7) => [3,3,1] => [3,1] => 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [4,3] => [3] => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [5,1,1] => [1,1] => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => [2,2,1] => 8
([(2,6),(3,5),(4,5),(4,6)],7) => [5,1,1] => [1,1] => 1
([(1,2),(3,6),(4,5),(5,6)],7) => [4,2,1] => [2,1] => 2
([(0,3),(1,2),(4,6),(5,6)],7) => [3,2,2] => [2,2] => 4
([(2,3),(4,5),(4,6),(5,6)],7) => [3,2,1,1] => [2,1,1] => 5
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [5,2] => [2] => 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1] => [2,1] => 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5,1,1] => [1,1] => 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => [4,2,1] => [2,1] => 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [4,3] => [3] => 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,3,1] => [3,1] => 3
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [5,2] => [2] => 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,3] => [3] => 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1] => [2,1] => 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [4,3] => [3] => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 3
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [5,2] => [2] => 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => [5,1,1] => [1,1] => 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => [4,3] => [3] => 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7) => [4,3] => [3] => 0
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,2,2] => [2,2] => 4
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5,2] => [2] => 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => [4,3] => [3] => 0
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3,1] => [3,1] => 3
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,2] => [2] => 0
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,3] => [3] => 0
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Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
Map
first row removal
Description
Removes the first entry of an integer partition
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