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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001142

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3

Description

The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001330

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => [2] => ([],2)
 => 1 = 0 + 1
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2] => 00 => [3] => ([],3)
 => 1 = 0 + 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,1] => 00 => [3] => ([],3)
 => 1 = 0 + 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,3,2] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,1,3] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,3,1] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,2,1] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[3,2,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,-2,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,-2,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-3,2,1] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-3,2,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[-3,-2,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-3,-2,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4] => 0000 => [5] => ([],5)
 => 1 = 0 + 1
[1,2,3,-4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,-3,4] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,-3,-4] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-2,3,4] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,-2,3,-4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-2,-3,4] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-2,-3,-4] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,2,3,-4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,2,-3,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,2,-3,-4] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,-2,3,-4] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,2,-4,-3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-2,4,-3] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-2,-4,3] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-2,-4,-3] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,2,4,-3] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,2,-4,3] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,2,-4,-3] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,-2,4,-3] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,3,-2,-4] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-3,2,-4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-3,-2,4] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-3,-2,-4] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,3,2,-4] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,3,-2,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,3,-2,-4] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,-3,2,-4] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,3,-4,-2] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-3,4,-2] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-3,-4,2] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-3,-4,-2] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,3,4,-2] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,3,-4,2] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,3,-4,-2] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,-3,4,-2] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,4,-2,-3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-4,2,-3] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-4,-2,3] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,-4,-2,-3] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[-1,4,2,-3] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[-1,4,-2,3] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001429
(load all 8 compositions to match this statistic)

Mapping

St001429: Signed permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%

Values

[1] => 0
[-1] => 1
[1,2] => 0
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 2
[2,1] => 0
[2,-1] => 1
[-2,1] => 1
[-2,-1] => 2
[1,2,3] => 0
[1,2,-3] => 1
[1,-2,3] => 1
[1,-2,-3] => 2
[-1,2,3] => 1
[-1,2,-3] => 2
[-1,-2,3] => 2
[-1,-2,-3] => 3
[1,3,2] => 0
[1,3,-2] => 1
[1,-3,2] => 1
[1,-3,-2] => 2
[-1,3,2] => 1
[-1,3,-2] => 2
[-1,-3,2] => 2
[-1,-3,-2] => 3
[2,1,3] => 0
[2,1,-3] => 1
[2,-1,3] => 1
[2,-1,-3] => 2
[-2,1,3] => 1
[-2,1,-3] => 2
[-2,-1,3] => 2
[-2,-1,-3] => 3
[2,3,1] => 0
[2,3,-1] => 1
[2,-3,1] => 1
[2,-3,-1] => 2
[-2,3,1] => 1
[-2,3,-1] => 2
[-2,-3,1] => 2
[-2,-3,-1] => 3
[3,1,2] => 0
[3,1,-2] => 1
[3,-1,2] => 1
[3,-1,-2] => 2
[-3,1,2] => 1
[-3,1,-2] => 2
[-3,-1,2] => 2
[-3,-1,-2] => 3
[1,2,3,-4,5] => ? = 1
[1,2,3,-4,-5] => ? = 2
[1,2,-3,4,5] => ? = 1
[1,2,-3,4,-5] => ? = 2
[1,2,-3,-4,5] => ? = 2
[1,2,-3,-4,-5] => ? = 3
[1,-2,3,4,5] => ? = 1
[1,-2,3,4,-5] => ? = 2
[1,-2,3,-4,5] => ? = 2
[1,-2,3,-4,-5] => ? = 3
[1,-2,-3,4,5] => ? = 2
[1,-2,-3,4,-5] => ? = 3
[1,-2,-3,-4,5] => ? = 3
[1,-2,-3,-4,-5] => ? = 4
[-1,2,3,4,-5] => ? = 2
[-1,2,3,-4,5] => ? = 2
[-1,2,3,-4,-5] => ? = 3
[-1,2,-3,4,5] => ? = 2
[-1,2,-3,4,-5] => ? = 2
[-1,2,-3,-4,5] => ? = 3
[-1,2,-3,-4,-5] => ? = 4
[-1,-2,3,4,5] => ? = 2
[-1,-2,3,4,-5] => ? = 3
[-1,-2,3,-4,5] => ? = 3
[-1,-2,3,-4,-5] => ? = 4
[-1,-2,-3,4,5] => ? = 3
[-1,-2,-3,4,-5] => ? = 4
[-1,-2,-3,-4,5] => ? = 4
[1,2,3,-5,-4] => ? = 2
[1,2,-3,5,4] => ? = 1
[1,2,-3,5,-4] => ? = 2
[1,2,-3,-5,4] => ? = 2
[1,2,-3,-5,-4] => ? = 3
[1,-2,3,5,4] => ? = 1
[1,-2,3,5,-4] => ? = 2
[1,-2,3,-5,4] => ? = 2
[1,-2,3,-5,-4] => ? = 3
[1,-2,-3,5,4] => ? = 2
[1,-2,-3,5,-4] => ? = 3
[1,-2,-3,-5,4] => ? = 3
[1,-2,-3,-5,-4] => ? = 4
[-1,2,3,5,-4] => ? = 2
[-1,2,3,-5,4] => ? = 2
[-1,2,3,-5,-4] => ? = 3
[-1,2,-3,5,4] => ? = 2
[-1,2,-3,5,-4] => ? = 2
[-1,2,-3,-5,4] => ? = 3
[-1,2,-3,-5,-4] => ? = 4
[-1,-2,3,5,4] => ? = 2
[-1,-2,3,5,-4] => ? = 3

Description

The number of negative entries in a signed permutation.

Matching statistic: St000939

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

[1] => []
 => ?
 => ?
 => ? = 0 - 2
[-1] => [1]
 => []
 => ?
 => ? = 1 - 2
[1,2] => []
 => ?
 => ?
 => ? = 0 - 2
[1,-2] => [1]
 => []
 => ?
 => ? = 1 - 2
[-1,2] => [1]
 => []
 => ?
 => ? = 1 - 2
[-1,-2] => [1,1]
 => [1]
 => []
 => ? = 2 - 2
[2,1] => []
 => ?
 => ?
 => ? = 0 - 2
[2,-1] => [2]
 => []
 => ?
 => ? = 1 - 2
[-2,1] => [2]
 => []
 => ?
 => ? = 1 - 2
[-2,-1] => []
 => ?
 => ?
 => ? = 2 - 2
[1,2,3] => []
 => ?
 => ?
 => ? = 0 - 2
[1,2,-3] => [1]
 => []
 => ?
 => ? = 1 - 2
[1,-2,3] => [1]
 => []
 => ?
 => ? = 1 - 2
[1,-2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,2,3] => [1]
 => []
 => ?
 => ? = 1 - 2
[-1,2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,-2,3] => [1,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 2
[1,3,2] => []
 => ?
 => ?
 => ? = 0 - 2
[1,3,-2] => [2]
 => []
 => ?
 => ? = 1 - 2
[1,-3,2] => [2]
 => []
 => ?
 => ? = 1 - 2
[1,-3,-2] => []
 => ?
 => ?
 => ? = 2 - 2
[-1,3,2] => [1]
 => []
 => ?
 => ? = 1 - 2
[-1,3,-2] => [2,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,-3,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,-3,-2] => [1]
 => []
 => ?
 => ? = 3 - 2
[2,1,3] => []
 => ?
 => ?
 => ? = 0 - 2
[2,1,-3] => [1]
 => []
 => ?
 => ? = 1 - 2
[2,-1,3] => [2]
 => []
 => ?
 => ? = 1 - 2
[2,-1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 2
[-2,1,3] => [2]
 => []
 => ?
 => ? = 1 - 2
[-2,1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 2
[-2,-1,3] => []
 => ?
 => ?
 => ? = 2 - 2
[-2,-1,-3] => [1]
 => []
 => ?
 => ? = 3 - 2
[2,3,1] => []
 => ?
 => ?
 => ? = 0 - 2
[2,3,-1] => [3]
 => []
 => ?
 => ? = 1 - 2
[2,-3,1] => [3]
 => []
 => ?
 => ? = 1 - 2
[2,-3,-1] => []
 => ?
 => ?
 => ? = 2 - 2
[-2,3,1] => [3]
 => []
 => ?
 => ? = 1 - 2
[-2,3,-1] => []
 => ?
 => ?
 => ? = 2 - 2
[-2,-3,1] => []
 => ?
 => ?
 => ? = 2 - 2
[-2,-3,-1] => [3]
 => []
 => ?
 => ? = 3 - 2
[3,1,2] => []
 => ?
 => ?
 => ? = 0 - 2
[3,1,-2] => [3]
 => []
 => ?
 => ? = 1 - 2
[3,-1,2] => [3]
 => []
 => ?
 => ? = 1 - 2
[3,-1,-2] => []
 => ?
 => ?
 => ? = 2 - 2
[-3,1,2] => [3]
 => []
 => ?
 => ? = 1 - 2
[-3,1,-2] => []
 => ?
 => ?
 => ? = 2 - 2
[-3,-1,2] => []
 => ?
 => ?
 => ? = 2 - 2
[-3,-1,-2] => [3]
 => []
 => ?
 => ? = 3 - 2
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 3 = 5 - 2
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,4,-3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-4,3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,5,-4,-3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-5,-4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,3,-2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-3,2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,4,-3,-2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-4,-3,2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,5,-3,-4,-2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-5,-3,-4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[2,-1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-2,1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[3,-2,-1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-3,-2,1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[4,-2,-3,-1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-4,-2,-3,1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[5,-2,-3,-4,-1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-5,-2,-3,-4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2

Description

The number of characters of the symmetric group whose value on the partition is positive.

Matching statistic: St000993

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

[1] => []
 => ?
 => ?
 => ? = 0 - 2
[-1] => [1]
 => []
 => ?
 => ? = 1 - 2
[1,2] => []
 => ?
 => ?
 => ? = 0 - 2
[1,-2] => [1]
 => []
 => ?
 => ? = 1 - 2
[-1,2] => [1]
 => []
 => ?
 => ? = 1 - 2
[-1,-2] => [1,1]
 => [1]
 => []
 => ? = 2 - 2
[2,1] => []
 => ?
 => ?
 => ? = 0 - 2
[2,-1] => [2]
 => []
 => ?
 => ? = 1 - 2
[-2,1] => [2]
 => []
 => ?
 => ? = 1 - 2
[-2,-1] => []
 => ?
 => ?
 => ? = 2 - 2
[1,2,3] => []
 => ?
 => ?
 => ? = 0 - 2
[1,2,-3] => [1]
 => []
 => ?
 => ? = 1 - 2
[1,-2,3] => [1]
 => []
 => ?
 => ? = 1 - 2
[1,-2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,2,3] => [1]
 => []
 => ?
 => ? = 1 - 2
[-1,2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,-2,3] => [1,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 2
[1,3,2] => []
 => ?
 => ?
 => ? = 0 - 2
[1,3,-2] => [2]
 => []
 => ?
 => ? = 1 - 2
[1,-3,2] => [2]
 => []
 => ?
 => ? = 1 - 2
[1,-3,-2] => []
 => ?
 => ?
 => ? = 2 - 2
[-1,3,2] => [1]
 => []
 => ?
 => ? = 1 - 2
[-1,3,-2] => [2,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,-3,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 2
[-1,-3,-2] => [1]
 => []
 => ?
 => ? = 3 - 2
[2,1,3] => []
 => ?
 => ?
 => ? = 0 - 2
[2,1,-3] => [1]
 => []
 => ?
 => ? = 1 - 2
[2,-1,3] => [2]
 => []
 => ?
 => ? = 1 - 2
[2,-1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 2
[-2,1,3] => [2]
 => []
 => ?
 => ? = 1 - 2
[-2,1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 2
[-2,-1,3] => []
 => ?
 => ?
 => ? = 2 - 2
[-2,-1,-3] => [1]
 => []
 => ?
 => ? = 3 - 2
[2,3,1] => []
 => ?
 => ?
 => ? = 0 - 2
[2,3,-1] => [3]
 => []
 => ?
 => ? = 1 - 2
[2,-3,1] => [3]
 => []
 => ?
 => ? = 1 - 2
[2,-3,-1] => []
 => ?
 => ?
 => ? = 2 - 2
[-2,3,1] => [3]
 => []
 => ?
 => ? = 1 - 2
[-2,3,-1] => []
 => ?
 => ?
 => ? = 2 - 2
[-2,-3,1] => []
 => ?
 => ?
 => ? = 2 - 2
[-2,-3,-1] => [3]
 => []
 => ?
 => ? = 3 - 2
[3,1,2] => []
 => ?
 => ?
 => ? = 0 - 2
[3,1,-2] => [3]
 => []
 => ?
 => ? = 1 - 2
[3,-1,2] => [3]
 => []
 => ?
 => ? = 1 - 2
[3,-1,-2] => []
 => ?
 => ?
 => ? = 2 - 2
[-3,1,2] => [3]
 => []
 => ?
 => ? = 1 - 2
[-3,1,-2] => []
 => ?
 => ?
 => ? = 2 - 2
[-3,-1,2] => []
 => ?
 => ?
 => ? = 2 - 2
[-3,-1,-2] => [3]
 => []
 => ?
 => ? = 3 - 2
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 3 = 5 - 2
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,4,-3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-4,3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,5,-4,-3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-2,-5,-4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,3,-2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-3,2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,4,-3,-2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-4,-3,2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,5,-3,-4,-2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-1,-5,-3,-4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[2,-1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-2,1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[3,-2,-1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-3,-2,1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[4,-2,-3,-1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-4,-2,-3,1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[5,-2,-3,-4,-1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2
[-5,-2,-3,-4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2 = 4 - 2

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St000510

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

[1] => []
 => ?
 => ?
 => ? = 0 - 3
[-1] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,2] => []
 => ?
 => ?
 => ? = 0 - 3
[1,-2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,-2] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[2,1] => []
 => ?
 => ?
 => ? = 0 - 3
[2,-1] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,1] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[1,2,3] => []
 => ?
 => ?
 => ? = 0 - 3
[1,2,-3] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,-2,3] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,-2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,2,3] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-2,3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 3
[1,3,2] => []
 => ?
 => ?
 => ? = 0 - 3
[1,3,-2] => [2]
 => []
 => ?
 => ? = 1 - 3
[1,-3,2] => [2]
 => []
 => ?
 => ? = 1 - 3
[1,-3,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-1,3,2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,3,-2] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-3,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-3,-2] => [1]
 => []
 => ?
 => ? = 3 - 3
[2,1,3] => []
 => ?
 => ?
 => ? = 0 - 3
[2,1,-3] => [1]
 => []
 => ?
 => ? = 1 - 3
[2,-1,3] => [2]
 => []
 => ?
 => ? = 1 - 3
[2,-1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-2,1,3] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-2,-1,3] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-1,-3] => [1]
 => []
 => ?
 => ? = 3 - 3
[2,3,1] => []
 => ?
 => ?
 => ? = 0 - 3
[2,3,-1] => [3]
 => []
 => ?
 => ? = 1 - 3
[2,-3,1] => [3]
 => []
 => ?
 => ? = 1 - 3
[2,-3,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,3,1] => [3]
 => []
 => ?
 => ? = 1 - 3
[-2,3,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-3,1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-3,-1] => [3]
 => []
 => ?
 => ? = 3 - 3
[3,1,2] => []
 => ?
 => ?
 => ? = 0 - 3
[3,1,-2] => [3]
 => []
 => ?
 => ? = 1 - 3
[3,-1,2] => [3]
 => []
 => ?
 => ? = 1 - 3
[3,-1,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,1,2] => [3]
 => []
 => ?
 => ? = 1 - 3
[-3,1,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,-1,2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,-1,-2] => [3]
 => []
 => ?
 => ? = 3 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2 = 5 - 3
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,4,-3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-4,3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,5,-4,-3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-5,-4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,3,-2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-3,2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,4,-3,-2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-4,-3,2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,5,-3,-4,-2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-5,-3,-4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[2,-1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-2,1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[3,-2,-1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-3,-2,1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[4,-2,-3,-1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-4,-2,-3,1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[5,-2,-3,-4,-1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-5,-2,-3,-4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3

Description

The number of invariant oriented cycles when acting with a permutation of given cycle type.

Matching statistic: St000681

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

[1] => []
 => ?
 => ?
 => ? = 0 - 3
[-1] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,2] => []
 => ?
 => ?
 => ? = 0 - 3
[1,-2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,-2] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[2,1] => []
 => ?
 => ?
 => ? = 0 - 3
[2,-1] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,1] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[1,2,3] => []
 => ?
 => ?
 => ? = 0 - 3
[1,2,-3] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,-2,3] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,-2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,2,3] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-2,3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 3
[1,3,2] => []
 => ?
 => ?
 => ? = 0 - 3
[1,3,-2] => [2]
 => []
 => ?
 => ? = 1 - 3
[1,-3,2] => [2]
 => []
 => ?
 => ? = 1 - 3
[1,-3,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-1,3,2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,3,-2] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-3,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-3,-2] => [1]
 => []
 => ?
 => ? = 3 - 3
[2,1,3] => []
 => ?
 => ?
 => ? = 0 - 3
[2,1,-3] => [1]
 => []
 => ?
 => ? = 1 - 3
[2,-1,3] => [2]
 => []
 => ?
 => ? = 1 - 3
[2,-1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-2,1,3] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-2,-1,3] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-1,-3] => [1]
 => []
 => ?
 => ? = 3 - 3
[2,3,1] => []
 => ?
 => ?
 => ? = 0 - 3
[2,3,-1] => [3]
 => []
 => ?
 => ? = 1 - 3
[2,-3,1] => [3]
 => []
 => ?
 => ? = 1 - 3
[2,-3,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,3,1] => [3]
 => []
 => ?
 => ? = 1 - 3
[-2,3,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-3,1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-3,-1] => [3]
 => []
 => ?
 => ? = 3 - 3
[3,1,2] => []
 => ?
 => ?
 => ? = 0 - 3
[3,1,-2] => [3]
 => []
 => ?
 => ? = 1 - 3
[3,-1,2] => [3]
 => []
 => ?
 => ? = 1 - 3
[3,-1,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,1,2] => [3]
 => []
 => ?
 => ? = 1 - 3
[-3,1,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,-1,2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,-1,-2] => [3]
 => []
 => ?
 => ? = 3 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2 = 5 - 3
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,4,-3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-4,3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,5,-4,-3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-5,-4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,3,-2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-3,2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,4,-3,-2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-4,-3,2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,5,-3,-4,-2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-5,-3,-4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[2,-1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-2,1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[3,-2,-1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-3,-2,1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[4,-2,-3,-1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-4,-2,-3,1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[5,-2,-3,-4,-1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-5,-2,-3,-4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3

Description

The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.

Matching statistic: St000937

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

[1] => []
 => ?
 => ?
 => ? = 0 - 3
[-1] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,2] => []
 => ?
 => ?
 => ? = 0 - 3
[1,-2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,-2] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[2,1] => []
 => ?
 => ?
 => ? = 0 - 3
[2,-1] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,1] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[1,2,3] => []
 => ?
 => ?
 => ? = 0 - 3
[1,2,-3] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,-2,3] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,-2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,2,3] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-2,3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 3
[1,3,2] => []
 => ?
 => ?
 => ? = 0 - 3
[1,3,-2] => [2]
 => []
 => ?
 => ? = 1 - 3
[1,-3,2] => [2]
 => []
 => ?
 => ? = 1 - 3
[1,-3,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-1,3,2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,3,-2] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-3,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-3,-2] => [1]
 => []
 => ?
 => ? = 3 - 3
[2,1,3] => []
 => ?
 => ?
 => ? = 0 - 3
[2,1,-3] => [1]
 => []
 => ?
 => ? = 1 - 3
[2,-1,3] => [2]
 => []
 => ?
 => ? = 1 - 3
[2,-1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-2,1,3] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-2,-1,3] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-1,-3] => [1]
 => []
 => ?
 => ? = 3 - 3
[2,3,1] => []
 => ?
 => ?
 => ? = 0 - 3
[2,3,-1] => [3]
 => []
 => ?
 => ? = 1 - 3
[2,-3,1] => [3]
 => []
 => ?
 => ? = 1 - 3
[2,-3,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,3,1] => [3]
 => []
 => ?
 => ? = 1 - 3
[-2,3,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-3,1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-3,-1] => [3]
 => []
 => ?
 => ? = 3 - 3
[3,1,2] => []
 => ?
 => ?
 => ? = 0 - 3
[3,1,-2] => [3]
 => []
 => ?
 => ? = 1 - 3
[3,-1,2] => [3]
 => []
 => ?
 => ? = 1 - 3
[3,-1,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,1,2] => [3]
 => []
 => ?
 => ? = 1 - 3
[-3,1,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,-1,2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,-1,-2] => [3]
 => []
 => ?
 => ? = 3 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2 = 5 - 3
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,4,-3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-4,3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,5,-4,-3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-5,-4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,3,-2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-3,2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,4,-3,-2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-4,-3,2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,5,-3,-4,-2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-5,-3,-4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[2,-1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-2,1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[3,-2,-1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-3,-2,1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[4,-2,-3,-1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-4,-2,-3,1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[5,-2,-3,-4,-1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-5,-2,-3,-4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3

Description

The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation

$S^{(2,2)}$ are

$2$ on the conjugacy classes

$(4)$ and

$(2,2)$ ,

$0$ on the conjugacy classes

$(3,1)$ and

$(1,1,1,1)$ , and

$-1$ on the conjugacy class

$(2,1,1)$ . Therefore, the statistic on the partition

$(2,2)$ is

$2$ .

Matching statistic: St001101

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001101: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

[1] => []
 => ?
 => ?
 => ? = 0 - 3
[-1] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,2] => []
 => ?
 => ?
 => ? = 0 - 3
[1,-2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,-2] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[2,1] => []
 => ?
 => ?
 => ? = 0 - 3
[2,-1] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,1] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[1,2,3] => []
 => ?
 => ?
 => ? = 0 - 3
[1,2,-3] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,-2,3] => [1]
 => []
 => ?
 => ? = 1 - 3
[1,-2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,2,3] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-2,3] => [1,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 3
[1,3,2] => []
 => ?
 => ?
 => ? = 0 - 3
[1,3,-2] => [2]
 => []
 => ?
 => ? = 1 - 3
[1,-3,2] => [2]
 => []
 => ?
 => ? = 1 - 3
[1,-3,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-1,3,2] => [1]
 => []
 => ?
 => ? = 1 - 3
[-1,3,-2] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-3,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-1,-3,-2] => [1]
 => []
 => ?
 => ? = 3 - 3
[2,1,3] => []
 => ?
 => ?
 => ? = 0 - 3
[2,1,-3] => [1]
 => []
 => ?
 => ? = 1 - 3
[2,-1,3] => [2]
 => []
 => ?
 => ? = 1 - 3
[2,-1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-2,1,3] => [2]
 => []
 => ?
 => ? = 1 - 3
[-2,1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 3
[-2,-1,3] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-1,-3] => [1]
 => []
 => ?
 => ? = 3 - 3
[2,3,1] => []
 => ?
 => ?
 => ? = 0 - 3
[2,3,-1] => [3]
 => []
 => ?
 => ? = 1 - 3
[2,-3,1] => [3]
 => []
 => ?
 => ? = 1 - 3
[2,-3,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,3,1] => [3]
 => []
 => ?
 => ? = 1 - 3
[-2,3,-1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-3,1] => []
 => ?
 => ?
 => ? = 2 - 3
[-2,-3,-1] => [3]
 => []
 => ?
 => ? = 3 - 3
[3,1,2] => []
 => ?
 => ?
 => ? = 0 - 3
[3,1,-2] => [3]
 => []
 => ?
 => ? = 1 - 3
[3,-1,2] => [3]
 => []
 => ?
 => ? = 1 - 3
[3,-1,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,1,2] => [3]
 => []
 => ?
 => ? = 1 - 3
[-3,1,-2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,-1,2] => []
 => ?
 => ?
 => ? = 2 - 3
[-3,-1,-2] => [3]
 => []
 => ?
 => ? = 3 - 3
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2 = 5 - 3
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,4,-3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-4,3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,5,-4,-3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-2,-5,-4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,3,-2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-3,2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,4,-3,-2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-4,-3,2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,5,-3,-4,-2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-1,-5,-3,-4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[2,-1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-2,1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[3,-2,-1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-3,-2,1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[4,-2,-3,-1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-4,-2,-3,1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[5,-2,-3,-4,-1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3
[-5,-2,-3,-4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 4 - 3

Description

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. For a generating function

$f$ the associated formal group law is the symmetric function

$f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$ , see [1]. This statistic records the coefficient of the monomial symmetric function

$m_\lambda$ times the product of the factorials of the parts of

$\lambda$ in the formal group law for increasing trees, whose generating function is

$f(x) = -\log(1-x)$ , see [1, sec. 9.1] Fix a coloring of

$\{1,2, \ldots, n\}$ so that

$\lambda_i$ are colored with the

$i$ th color. This statistic gives the number of increasing trees on this colored set of vertices so that no leaf has the same color as its parent. (An increasing tree is a rooted tree on the vertex set

$\{1,2, \ldots, n\}$ with the property that any child of

$i$ is greater than

$i$ .)

Matching statistic: St000512

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%

Values

[1] => []
 => ?
 => ?
 => ? = 0 - 4
[-1] => [1]
 => []
 => ?
 => ? = 1 - 4
[1,2] => []
 => ?
 => ?
 => ? = 0 - 4
[1,-2] => [1]
 => []
 => ?
 => ? = 1 - 4
[-1,2] => [1]
 => []
 => ?
 => ? = 1 - 4
[-1,-2] => [1,1]
 => [1]
 => []
 => ? = 2 - 4
[2,1] => []
 => ?
 => ?
 => ? = 0 - 4
[2,-1] => [2]
 => []
 => ?
 => ? = 1 - 4
[-2,1] => [2]
 => []
 => ?
 => ? = 1 - 4
[-2,-1] => []
 => ?
 => ?
 => ? = 2 - 4
[1,2,3] => []
 => ?
 => ?
 => ? = 0 - 4
[1,2,-3] => [1]
 => []
 => ?
 => ? = 1 - 4
[1,-2,3] => [1]
 => []
 => ?
 => ? = 1 - 4
[1,-2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 4
[-1,2,3] => [1]
 => []
 => ?
 => ? = 1 - 4
[-1,2,-3] => [1,1]
 => [1]
 => []
 => ? = 2 - 4
[-1,-2,3] => [1,1]
 => [1]
 => []
 => ? = 2 - 4
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 4
[1,3,2] => []
 => ?
 => ?
 => ? = 0 - 4
[1,3,-2] => [2]
 => []
 => ?
 => ? = 1 - 4
[1,-3,2] => [2]
 => []
 => ?
 => ? = 1 - 4
[1,-3,-2] => []
 => ?
 => ?
 => ? = 2 - 4
[-1,3,2] => [1]
 => []
 => ?
 => ? = 1 - 4
[-1,3,-2] => [2,1]
 => [1]
 => []
 => ? = 2 - 4
[-1,-3,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 4
[-1,-3,-2] => [1]
 => []
 => ?
 => ? = 3 - 4
[2,1,3] => []
 => ?
 => ?
 => ? = 0 - 4
[2,1,-3] => [1]
 => []
 => ?
 => ? = 1 - 4
[2,-1,3] => [2]
 => []
 => ?
 => ? = 1 - 4
[2,-1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 4
[-2,1,3] => [2]
 => []
 => ?
 => ? = 1 - 4
[-2,1,-3] => [2,1]
 => [1]
 => []
 => ? = 2 - 4
[-2,-1,3] => []
 => ?
 => ?
 => ? = 2 - 4
[-2,-1,-3] => [1]
 => []
 => ?
 => ? = 3 - 4
[2,3,1] => []
 => ?
 => ?
 => ? = 0 - 4
[2,3,-1] => [3]
 => []
 => ?
 => ? = 1 - 4
[2,-3,1] => [3]
 => []
 => ?
 => ? = 1 - 4
[2,-3,-1] => []
 => ?
 => ?
 => ? = 2 - 4
[-2,3,1] => [3]
 => []
 => ?
 => ? = 1 - 4
[-2,3,-1] => []
 => ?
 => ?
 => ? = 2 - 4
[-2,-3,1] => []
 => ?
 => ?
 => ? = 2 - 4
[-2,-3,-1] => [3]
 => []
 => ?
 => ? = 3 - 4
[3,1,2] => []
 => ?
 => ?
 => ? = 0 - 4
[3,1,-2] => [3]
 => []
 => ?
 => ? = 1 - 4
[3,-1,2] => [3]
 => []
 => ?
 => ? = 1 - 4
[3,-1,-2] => []
 => ?
 => ?
 => ? = 2 - 4
[-3,1,2] => [3]
 => []
 => ?
 => ? = 1 - 4
[-3,1,-2] => []
 => ?
 => ?
 => ? = 2 - 4
[-3,-1,2] => []
 => ?
 => ?
 => ? = 2 - 4
[-3,-1,-2] => [3]
 => []
 => ?
 => ? = 3 - 4
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 5 - 4
[-1,-2,-3,5,-4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,-3,-5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,4,-3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,-4,3,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,5,-4,-3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-2,-5,-4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,3,-2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-3,2,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,4,-3,-2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-4,-3,2,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,5,-3,-4,-2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-1,-5,-3,-4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[2,-1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-2,1,-3,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[3,-2,-1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-3,-2,1,-4,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[4,-2,-3,-1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-4,-2,-3,1,-5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[5,-2,-3,-4,-1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4
[-5,-2,-3,-4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 4 - 4

Description

The number of invariant subsets of size 3 when acting with a permutation of given cycle type.

The following 1 statistic also match your data. Click on any of them to see the details.

St000941The number of characters of the symmetric group whose value on the partition is even.