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Your data matches 101 different statistics following compositions of up to 3 maps.
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Matching statistic: St000221

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of strong fixed points of a permutation.

$i$ is called a strong fixed point of

$\pi$ if 1.

$j < i$ implies

$\pi_j < \pi_i$ , and 2.

$j > i$ implies

$\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

Matching statistic: St000627

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[-1] => [1]
 => 10 => 00 => 2 = 1 + 1
[1,-2] => [1]
 => 10 => 00 => 2 = 1 + 1
[-1,2] => [1]
 => 10 => 00 => 2 = 1 + 1
[-1,-2] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[2,-1] => [2]
 => 100 => 000 => 3 = 2 + 1
[-2,1] => [2]
 => 100 => 000 => 3 = 2 + 1
[1,2,-3] => [1]
 => 10 => 00 => 2 = 1 + 1
[1,-2,3] => [1]
 => 10 => 00 => 2 = 1 + 1
[1,-2,-3] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[-1,2,3] => [1]
 => 10 => 00 => 2 = 1 + 1
[-1,2,-3] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[-1,-2,3] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[-1,-2,-3] => [1,1,1]
 => 1110 => 0110 => 1 = 0 + 1
[1,3,-2] => [2]
 => 100 => 000 => 3 = 2 + 1
[1,-3,2] => [2]
 => 100 => 000 => 3 = 2 + 1
[-1,3,2] => [1]
 => 10 => 00 => 2 = 1 + 1
[-1,3,-2] => [2,1]
 => 1010 => 0010 => 1 = 0 + 1
[-1,-3,2] => [2,1]
 => 1010 => 0010 => 1 = 0 + 1
[-1,-3,-2] => [1]
 => 10 => 00 => 2 = 1 + 1
[2,1,-3] => [1]
 => 10 => 00 => 2 = 1 + 1
[2,-1,3] => [2]
 => 100 => 000 => 3 = 2 + 1
[2,-1,-3] => [2,1]
 => 1010 => 0010 => 1 = 0 + 1
[-2,1,3] => [2]
 => 100 => 000 => 3 = 2 + 1
[-2,1,-3] => [2,1]
 => 1010 => 0010 => 1 = 0 + 1
[-2,-1,-3] => [1]
 => 10 => 00 => 2 = 1 + 1
[2,3,-1] => [3]
 => 1000 => 0000 => 4 = 3 + 1
[2,-3,1] => [3]
 => 1000 => 0000 => 4 = 3 + 1
[-2,3,1] => [3]
 => 1000 => 0000 => 4 = 3 + 1
[-2,-3,-1] => [3]
 => 1000 => 0000 => 4 = 3 + 1
[3,1,-2] => [3]
 => 1000 => 0000 => 4 = 3 + 1
[3,-1,2] => [3]
 => 1000 => 0000 => 4 = 3 + 1
[-3,1,2] => [3]
 => 1000 => 0000 => 4 = 3 + 1
[-3,-1,-2] => [3]
 => 1000 => 0000 => 4 = 3 + 1
[3,2,-1] => [2]
 => 100 => 000 => 3 = 2 + 1
[3,-2,1] => [1]
 => 10 => 00 => 2 = 1 + 1
[3,-2,-1] => [2,1]
 => 1010 => 0010 => 1 = 0 + 1
[-3,2,1] => [2]
 => 100 => 000 => 3 = 2 + 1
[-3,-2,1] => [2,1]
 => 1010 => 0010 => 1 = 0 + 1
[-3,-2,-1] => [1]
 => 10 => 00 => 2 = 1 + 1
[1,2,3,-4] => [1]
 => 10 => 00 => 2 = 1 + 1
[1,2,-3,4] => [1]
 => 10 => 00 => 2 = 1 + 1
[1,2,-3,-4] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[1,-2,3,4] => [1]
 => 10 => 00 => 2 = 1 + 1
[1,-2,3,-4] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[1,-2,-3,4] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[1,-2,-3,-4] => [1,1,1]
 => 1110 => 0110 => 1 = 0 + 1
[-1,2,3,4] => [1]
 => 10 => 00 => 2 = 1 + 1
[-1,2,3,-4] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[-1,2,-3,4] => [1,1]
 => 110 => 010 => 1 = 0 + 1
[-1,2,-3,-4] => [1,1,1]
 => 1110 => 0110 => 1 = 0 + 1

Description

The exponent of a binary word. This is the largest number

$e$ such that

$w$ is the concatenation of

$e$ identical factors. This statistic is also called '''frequency'''.

Matching statistic: St000968

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000968: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[-1] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,-2] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[-1,2] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,-1] => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[-2,1] => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[1,2,-3] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,-2,3] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[-1,2,3] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[-1,3,2] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[-1,-3,-2] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[2,1,-3] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[-2,-1,-3] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[3,2,-1] => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[3,-2,1] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[3,-2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[-3,2,1] => [2]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[-3,-2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[-3,-2,-1] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,2,3,-4] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,2,-3,4] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,-2,3,4] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,-2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[-1,2,3,4] => [1]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[-1,2,-3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1

Description

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . Then we calculate the dominant dimension of that CNakayama algebra.

Matching statistic: St001006
(load all 2 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001006: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%

Values

[-1] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,2] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[2,-1] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-2,1] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,-3] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,3] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[-1,2,3] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[-1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[-1,-2,-3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,-2] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,-3,2] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,3,2] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[-1,-3,-2] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,1,-3] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,-1,3] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[-2,1,3] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[-2,-1,-3] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,-3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[-2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[-2,-3,-1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,-1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[-3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[-3,-1,-2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,2,-1] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,-2,1] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[3,-2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[-3,2,1] => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-3,-2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[-3,-2,-1] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,2,3,-4] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,2,-3,4] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,-2,3,4] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,-2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[-1,2,3,4] => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[-1,2,-3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[-6,-4,-2,1,3,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,2,3,-6,1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,-6,-4,-5,1,3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-5,3,6,1,2,4] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,-5,-6,-3,1,2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-4,3,-5,2,-6,1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[6,1,2,3,4,-5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[5,1,2,3,6,-4] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[5,1,2,6,4,-3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,1,2,5,6,-3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[5,1,6,3,4,-2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,1,5,3,6,-2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[5,1,6,2,3,-4] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,1,5,6,4,-2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,1,5,6,2,-3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,1,4,5,6,-2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,2,3,4,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,2,3,6,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,5,2,6,4,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,4,2,5,6,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,1,2,4,-3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,1,2,6,-3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,1,6,3,-2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,5,6,3,4,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,4,5,3,6,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,6,2,3,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,5,6,1,4,-2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,4,5,1,6,-2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,3,5,6,4,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,4,5,6,2,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,3,4,5,6,-1] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,6,-4,1,3,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,-6,1,3,4,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[2,3,-6,1,4,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-6,3,4,1,2,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,6,-3,1,2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,6,-5,1,2,4] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-6,3,1,2,4,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[6,-3,1,2,4,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[5,6,-4,1,2,3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-4,6,1,2,3,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,-6,1,2,3,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[6,-5,1,2,3,4] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-6,1,2,3,4,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[3,1,-6,2,4,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-2,-5,-6,1,3,4] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,2,-6,1,3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[-3,-4,2,-6,1,5] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,6,3,1,-2] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1
[4,5,2,6,1,-3] => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6 + 1

Description

Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001107
(load all 4 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 86%

Values

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

Description

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

Matching statistic: St000205
(load all 7 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 35%●distinct values known / distinct values provided: 14%

Values

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

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given

$\lambda$ count how many ''integer partitions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex.

Matching statistic: St000206
(load all 7 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 35%●distinct values known / distinct values provided: 14%

Values

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

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given

$\lambda$ count how many ''integer compositions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].

Matching statistic: St000781
(load all 7 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 35%●distinct values known / distinct values provided: 14%

Values

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

Description

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

Matching statistic: St001901
(load all 7 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 35%●distinct values known / distinct values provided: 14%

Values

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

Description

The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.

Matching statistic: St000290

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 35%●distinct values known / distinct values provided: 14%

Values

[-1] => [1]
 => []
 => ? => ? = 1
[1,-2] => [1]
 => []
 => ? => ? = 1
[-1,2] => [1]
 => []
 => ? => ? = 1
[-1,-2] => [1,1]
 => [1]
 => 1 => 0
[2,-1] => [2]
 => []
 => ? => ? = 2
[-2,1] => [2]
 => []
 => ? => ? = 2
[1,2,-3] => [1]
 => []
 => ? => ? = 1
[1,-2,3] => [1]
 => []
 => ? => ? = 1
[1,-2,-3] => [1,1]
 => [1]
 => 1 => 0
[-1,2,3] => [1]
 => []
 => ? => ? = 1
[-1,2,-3] => [1,1]
 => [1]
 => 1 => 0
[-1,-2,3] => [1,1]
 => [1]
 => 1 => 0
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => 11 => 0
[1,3,-2] => [2]
 => []
 => ? => ? = 2
[1,-3,2] => [2]
 => []
 => ? => ? = 2
[-1,3,2] => [1]
 => []
 => ? => ? = 1
[-1,3,-2] => [2,1]
 => [1]
 => 1 => 0
[-1,-3,2] => [2,1]
 => [1]
 => 1 => 0
[-1,-3,-2] => [1]
 => []
 => ? => ? = 1
[2,1,-3] => [1]
 => []
 => ? => ? = 1
[2,-1,3] => [2]
 => []
 => ? => ? = 2
[2,-1,-3] => [2,1]
 => [1]
 => 1 => 0
[-2,1,3] => [2]
 => []
 => ? => ? = 2
[-2,1,-3] => [2,1]
 => [1]
 => 1 => 0
[-2,-1,-3] => [1]
 => []
 => ? => ? = 1
[2,3,-1] => [3]
 => []
 => ? => ? = 3
[2,-3,1] => [3]
 => []
 => ? => ? = 3
[-2,3,1] => [3]
 => []
 => ? => ? = 3
[-2,-3,-1] => [3]
 => []
 => ? => ? = 3
[3,1,-2] => [3]
 => []
 => ? => ? = 3
[3,-1,2] => [3]
 => []
 => ? => ? = 3
[-3,1,2] => [3]
 => []
 => ? => ? = 3
[-3,-1,-2] => [3]
 => []
 => ? => ? = 3
[3,2,-1] => [2]
 => []
 => ? => ? = 2
[3,-2,1] => [1]
 => []
 => ? => ? = 1
[3,-2,-1] => [2,1]
 => [1]
 => 1 => 0
[-3,2,1] => [2]
 => []
 => ? => ? = 2
[-3,-2,1] => [2,1]
 => [1]
 => 1 => 0
[-3,-2,-1] => [1]
 => []
 => ? => ? = 1
[1,2,3,-4] => [1]
 => []
 => ? => ? = 1
[1,2,-3,4] => [1]
 => []
 => ? => ? = 1
[1,2,-3,-4] => [1,1]
 => [1]
 => 1 => 0
[1,-2,3,4] => [1]
 => []
 => ? => ? = 1
[1,-2,3,-4] => [1,1]
 => [1]
 => 1 => 0
[1,-2,-3,4] => [1,1]
 => [1]
 => 1 => 0
[1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 11 => 0
[-1,2,3,4] => [1]
 => []
 => ? => ? = 1
[-1,2,3,-4] => [1,1]
 => [1]
 => 1 => 0
[-1,2,-3,4] => [1,1]
 => [1]
 => 1 => 0
[-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 11 => 0
[-1,-2,3,4] => [1,1]
 => [1]
 => 1 => 0
[-1,-2,3,-4] => [1,1,1]
 => [1,1]
 => 11 => 0
[-1,-2,-3,4] => [1,1,1]
 => [1,1]
 => 11 => 0
[-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 111 => 0
[1,2,4,-3] => [2]
 => []
 => ? => ? = 2
[1,2,-4,3] => [2]
 => []
 => ? => ? = 2
[1,-2,4,3] => [1]
 => []
 => ? => ? = 1
[1,-2,4,-3] => [2,1]
 => [1]
 => 1 => 0
[1,-2,-4,3] => [2,1]
 => [1]
 => 1 => 0
[1,-2,-4,-3] => [1]
 => []
 => ? => ? = 1
[-1,2,4,3] => [1]
 => []
 => ? => ? = 1
[-1,2,4,-3] => [2,1]
 => [1]
 => 1 => 0
[-1,2,-4,3] => [2,1]
 => [1]
 => 1 => 0
[-1,2,-4,-3] => [1]
 => []
 => ? => ? = 1
[-1,-2,4,3] => [1,1]
 => [1]
 => 1 => 0
[-1,-2,4,-3] => [2,1,1]
 => [1,1]
 => 11 => 0
[-1,-2,-4,3] => [2,1,1]
 => [1,1]
 => 11 => 0
[-1,-2,-4,-3] => [1,1]
 => [1]
 => 1 => 0
[1,3,2,-4] => [1]
 => []
 => ? => ? = 1
[1,3,-2,4] => [2]
 => []
 => ? => ? = 2
[1,3,-2,-4] => [2,1]
 => [1]
 => 1 => 0
[1,-3,2,4] => [2]
 => []
 => ? => ? = 2
[1,-3,2,-4] => [2,1]
 => [1]
 => 1 => 0
[1,-3,-2,-4] => [1]
 => []
 => ? => ? = 1
[-1,3,2,4] => [1]
 => []
 => ? => ? = 1
[-1,3,2,-4] => [1,1]
 => [1]
 => 1 => 0
[-1,3,-2,4] => [2,1]
 => [1]
 => 1 => 0
[-1,3,-2,-4] => [2,1,1]
 => [1,1]
 => 11 => 0
[-1,-3,2,4] => [2,1]
 => [1]
 => 1 => 0
[-1,-3,2,-4] => [2,1,1]
 => [1,1]
 => 11 => 0
[-1,-3,-2,4] => [1]
 => []
 => ? => ? = 1
[-1,-3,-2,-4] => [1,1]
 => [1]
 => 1 => 0
[1,3,4,-2] => [3]
 => []
 => ? => ? = 3
[1,3,-4,2] => [3]
 => []
 => ? => ? = 3
[1,-3,4,2] => [3]
 => []
 => ? => ? = 3
[1,-3,-4,-2] => [3]
 => []
 => ? => ? = 3
[-1,3,4,2] => [1]
 => []
 => ? => ? = 1
[-1,3,4,-2] => [3,1]
 => [1]
 => 1 => 0
[-1,3,-4,2] => [3,1]
 => [1]
 => 1 => 0
[-1,3,-4,-2] => [1]
 => []
 => ? => ? = 1
[-1,-3,4,2] => [3,1]
 => [1]
 => 1 => 0
[-1,-3,-4,-2] => [3,1]
 => [1]
 => 1 => 0
[-1,4,2,-3] => [3,1]
 => [1]
 => 1 => 0
[-1,4,-2,3] => [3,1]
 => [1]
 => 1 => 0
[-1,-4,2,3] => [3,1]
 => [1]
 => 1 => 0
[-1,-4,-2,-3] => [3,1]
 => [1]
 => 1 => 0
[1,4,-3,-2] => [2,1]
 => [1]
 => 1 => 0
[1,-4,-3,2] => [2,1]
 => [1]
 => 1 => 0
[-1,4,3,-2] => [2,1]
 => [1]
 => 1 => 0
[-1,4,-3,2] => [1,1]
 => [1]
 => 1 => 0

Description

The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length

$n$ with

$a$ zeros, the generating function for the major index is the

$q$ -binomial coefficient

$\binom{n}{a}_q$ .

The following 91 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000517The Kreweras number of an integer partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000630The length of the shortest palindromic decomposition of a binary word. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001471The magnitude of a Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition.