- FindStat

Your data matches 36 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000459
(load all 2 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.

Matching statistic: St001004
(load all 3 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => []
 => []
 => [] => 0
[-1] => [1]
 => [1,0]
 => [1] => 1
[1,2] => []
 => []
 => [] => 0
[1,-2] => [1]
 => [1,0]
 => [1] => 1
[-1,2] => [1]
 => [1,0]
 => [1] => 1
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[2,1] => []
 => []
 => [] => 0
[2,-1] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[-2,1] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[-2,-1] => []
 => []
 => [] => 0
[1,2,3] => []
 => []
 => [] => 0
[1,2,-3] => [1]
 => [1,0]
 => [1] => 1
[1,-2,3] => [1]
 => [1,0]
 => [1] => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,2,3] => [1]
 => [1,0]
 => [1] => 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[1,3,2] => []
 => []
 => [] => 0
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[1,-3,-2] => []
 => []
 => [] => 0
[-1,3,2] => [1]
 => [1,0]
 => [1] => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-1,-3,-2] => [1]
 => [1,0]
 => [1] => 1
[2,1,3] => []
 => []
 => [] => 0
[2,1,-3] => [1]
 => [1,0]
 => [1] => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,2] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 3
[-2,-1,3] => []
 => []
 => [] => 0
[-2,-1,-3] => [1]
 => [1,0]
 => [1] => 1
[2,3,1] => []
 => []
 => [] => 0
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[2,-3,-1] => []
 => []
 => [] => 0
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-2,3,-1] => []
 => []
 => [] => 0
[-2,-3,1] => []
 => []
 => [] => 0
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[3,1,2] => []
 => []
 => [] => 0
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[3,-1,-2] => []
 => []
 => [] => 0
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-3,1,-2] => []
 => []
 => [] => 0
[-3,-1,2] => []
 => []
 => [] => 0
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[-8,-6,-3,-7,-5,1,2,4] => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => ? = 8
[-6,-2,-3,1,7,8,4,5] => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => ? = 8

Description

The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a

$321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].

Matching statistic: St000019

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1] => []
 => []
 => [1] => 0
[-1] => [1]
 => [1,0]
 => [2,1] => 1
[1,2] => []
 => []
 => [1] => 0
[1,-2] => [1]
 => [1,0]
 => [2,1] => 1
[-1,2] => [1]
 => [1,0]
 => [2,1] => 1
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[2,1] => []
 => []
 => [1] => 0
[2,-1] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[-2,1] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[-2,-1] => []
 => []
 => [1] => 0
[1,2,3] => []
 => []
 => [1] => 0
[1,2,-3] => [1]
 => [1,0]
 => [2,1] => 1
[1,-2,3] => [1]
 => [1,0]
 => [2,1] => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[-1,2,3] => [1]
 => [1,0]
 => [2,1] => 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 3
[1,3,2] => []
 => []
 => [1] => 0
[1,3,-2] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[1,-3,-2] => []
 => []
 => [1] => 0
[-1,3,2] => [1]
 => [1,0]
 => [2,1] => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[-1,-3,-2] => [1]
 => [1,0]
 => [2,1] => 1
[2,1,3] => []
 => []
 => [1] => 0
[2,1,-3] => [1]
 => [1,0]
 => [2,1] => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[-2,1,3] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[-2,-1,3] => []
 => []
 => [1] => 0
[-2,-1,-3] => [1]
 => [1,0]
 => [2,1] => 1
[2,3,1] => []
 => []
 => [1] => 0
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,-3,-1] => []
 => []
 => [1] => 0
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-2,3,-1] => []
 => []
 => [1] => 0
[-2,-3,1] => []
 => []
 => [1] => 0
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,1,2] => []
 => []
 => [1] => 0
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,-1,-2] => []
 => []
 => [1] => 0
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-3,1,-2] => []
 => []
 => [1] => 0
[-3,-1,2] => []
 => []
 => [1] => 0
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-5,4,-8,-6,-2,1,3,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[4,-5,-8,-6,-3,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[3,-8,-5,-6,-4,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => ? = 6
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-6,-2,-3,1,4,5] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[4,1,-7,-6,-5,2,3] => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ? = 6
[-6,1,-3,2,-5,4] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[3,-2,5,-4,-6,1] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[3,-2,-4,6,-5,1] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[-2,4,-3,6,-5,1] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[4,8,-7,-6,3,1,2,5] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[4,-8,6,-7,-5,-3,1,2] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[-7,-3,2,8,-5,1,4,6] => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,1,2,3,8,7,4,6] => ? = 7
[-8,-3,2,6,-7,-5,1,4] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[7,8,-6,2,3,5,-4,1] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[-8,-6,-3,-7,-5,1,2,4] => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [9,1,2,3,4,5,8,6,7] => ? = 8
[4,7,8,-6,2,3,-5,1] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[6,-2,-4,1,-5,3] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[4,-6,-3,2,-5,1] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[2,-4,-3,1,-6,5,-8,7] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[2,-6,4,-3,-5,1,-8,7] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[2,-8,4,-3,6,-5,-7,1] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[5,6,2,3,-4,1,-8,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[5,6,1,-3,2,4,-8,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-2,1,4,-6,-5,3,-8,7] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[4,-6,1,-3,-5,2,-8,7] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[-2,1,4,-8,6,-5,-7,3] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[4,-8,1,-3,6,-5,-7,2] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[3,5,-8,2,-7,-4,-6,1] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[3,8,-7,-6,-5,2,1,4] => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => ? = 6
[7,8,-3,-6,2,4,-5,1] => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,1,2,3,8,7,4,6] => ? = 7
[7,8,2,3,-6,5,-4,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-2,1,-4,3,6,-8,-7,5] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[2,-4,-3,1,6,-8,-7,5] => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => ? = 6
[-2,1,6,-8,3,-5,-7,4] => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ? = 6
[7,8,5,1,-6,2,-4,3] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[7,8,4,-6,2,3,-5,1] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[2,8,-7,-6,-5,4,1,3] => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,1,2,3,8,7,4,6] => ? = 7
[7,8,1,-3,-6,5,2,4] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[5,3,-8,1,-7,-4,-6,2] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[2,-4,-3,6,-5,1] => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 6
[-2,-3,-4,1,-6,5,-8,7] => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 6

Description

The cardinality of the support of a permutation. A permutation

$\sigma$ may be written as a product

$\sigma = s_{i_1}\dots s_{i_k}$ with

$k$ minimal, where

$s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions

$i$ and

$i+1$ . The set of indices

$\{i_1,\dots,i_k\}$ is the '''support''' of

$\sigma$ and independent of the chosen way to write

$\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of

$\sigma$ of length

$n$ is the set of indices

$1 \leq i < n$ such that

$\sigma(k) < i$ for all

$k < i$ . Thus, the connectivity set is the complement of the support.

Matching statistic: St000026

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 97%●distinct values known / distinct values provided: 78%

Values

[1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-1] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[1,-2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[-2,-1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[1,2,3] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[1,2,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,2,3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,3,2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,-3,-2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-1,3,2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[-1,-3,-2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,1,3] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[2,1,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[-2,-1,3] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-2,-1,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,3,1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,-3,-1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[-2,3,-1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-2,-3,1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,1,2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,-1,-2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[-3,1,-2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-3,-1,2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[3,4,5,6,7,1,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[3,4,6,1,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[4,1,5,6,7,3,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[4,1,6,3,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[5,6,2,7,4,1,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[6,1,2,-7,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[-7,4,-3,1,2,5,6] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[6,1,7,-5,2,3,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[7,-5,-4,-6,1,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[-2,-5,-7,1,3,4,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[7,-6,2,1,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[-3,-5,7,-6,1,2,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[5,4,1,-7,2,3,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[7,4,1,-6,2,3,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[-5,4,1,-7,-6,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[3,1,6,2,4,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[3,1,5,2,8,4,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[4,5,1,2,8,3,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[4,6,1,2,3,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[6,1,5,2,4,8,7,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[6,5,1,2,3,8,7,-4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[7,4,1,8,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[1,7,5,3,8,4,6,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[8,1,5,2,7,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[8,5,1,2,7,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[8,4,1,6,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[8,6,7,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[8,6,1,7,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1

Description

The position of the first return of a Dyck path.

Matching statistic: St001348
(load all 3 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 97%●distinct values known / distinct values provided: 78%

Values

[1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-1] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[1,-2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[-2,-1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[1,2,3] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[1,2,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,2,3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,3,2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,-3,-2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-1,3,2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[-1,-3,-2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,1,3] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[2,1,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[-2,-1,3] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-2,-1,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2,3,1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,-3,-1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[-2,3,-1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-2,-3,1] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,1,2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,-1,-2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[-3,1,-2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-3,-1,2] => []
 => []
 => [1,0]
 => 1 = 0 + 1
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 1
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[3,4,5,6,7,1,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[3,4,6,1,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[4,1,5,6,7,3,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[4,1,6,3,7,5,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[5,6,2,7,4,1,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[6,1,2,-7,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[-7,4,-3,1,2,5,6] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[6,1,7,-5,2,3,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[7,-5,-4,-6,1,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[-2,-5,-7,1,3,4,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[7,-6,2,1,3,4,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[-3,-5,7,-6,1,2,4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[5,4,1,-7,2,3,6] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[7,4,1,-6,2,3,5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[-5,4,1,-7,-6,2,3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[3,1,6,2,4,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[3,1,5,2,8,4,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[4,5,1,2,8,3,6,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[4,6,1,2,3,8,7,-5] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[6,1,5,2,4,8,7,-3] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[6,5,1,2,3,8,7,-4] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[7,4,1,8,2,3,5,-6] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[1,7,5,3,8,4,6,-2] => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[8,1,5,2,7,4,6,-3] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[8,5,1,2,7,3,6,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[8,4,1,6,2,3,5,-7] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[8,6,7,1,2,3,4,-5] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1
[8,6,1,7,2,3,5,-4] => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 8 + 1

Description

The bounce of the parallelogram polyomino associated with the Dyck path. A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].

Matching statistic: St001958

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%

Values

[1] => []
 => []
 => [1] => 0
[-1] => [1]
 => [1,0]
 => [2,1] => 1
[1,2] => []
 => []
 => [1] => 0
[1,-2] => [1]
 => [1,0]
 => [2,1] => 1
[-1,2] => [1]
 => [1,0]
 => [2,1] => 1
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[2,1] => []
 => []
 => [1] => 0
[2,-1] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[-2,1] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[-2,-1] => []
 => []
 => [1] => 0
[1,2,3] => []
 => []
 => [1] => 0
[1,2,-3] => [1]
 => [1,0]
 => [2,1] => 1
[1,-2,3] => [1]
 => [1,0]
 => [2,1] => 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[-1,2,3] => [1]
 => [1,0]
 => [2,1] => 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 3
[1,3,2] => []
 => []
 => [1] => 0
[1,3,-2] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[1,-3,-2] => []
 => []
 => [1] => 0
[-1,3,2] => [1]
 => [1,0]
 => [2,1] => 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[-1,-3,-2] => [1]
 => [1,0]
 => [2,1] => 1
[2,1,3] => []
 => []
 => [1] => 0
[2,1,-3] => [1]
 => [1,0]
 => [2,1] => 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[-2,1,3] => [2]
 => [1,0,1,0]
 => [3,1,2] => 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
[-2,-1,3] => []
 => []
 => [1] => 0
[-2,-1,-3] => [1]
 => [1,0]
 => [2,1] => 1
[2,3,1] => []
 => []
 => [1] => 0
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[2,-3,-1] => []
 => []
 => [1] => 0
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-2,3,-1] => []
 => []
 => [1] => 0
[-2,-3,1] => []
 => []
 => [1] => 0
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,1,2] => []
 => []
 => [1] => 0
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[3,-1,-2] => []
 => []
 => [1] => 0
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-3,1,-2] => []
 => []
 => [1] => 0
[-3,-1,2] => []
 => []
 => [1] => 0
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3
[-6,-4,-2,1,3,5] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[4,5,2,3,-6,1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[2,-6,-4,-5,1,3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[-5,3,6,1,2,4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[4,-5,-6,-3,1,2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[-4,3,-5,2,-6,1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 7
[5,-3,2,4,8,-7,1,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-5,4,-8,-6,-2,1,3,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[4,-5,-8,-6,-3,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[3,-8,-5,-6,-4,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[-7,-6,3,-8,4,-5,1,2] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[4,2,-7,8,-6,1,3,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => ? = 6
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[-7,2,-4,6,-8,-5,1,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 7
[5,7,-6,4,2,3,-8,1] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 7
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 7
[6,1,2,3,4,-5] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[5,1,2,3,6,-4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[5,1,2,6,4,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[4,1,2,5,6,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[5,1,6,3,4,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[4,1,5,3,6,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[5,1,6,2,3,-4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[3,1,5,6,4,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[4,1,5,6,2,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[3,1,4,5,6,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[5,6,2,3,4,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[4,5,2,3,6,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6
[3,5,2,6,4,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6

Description

The degree of the polynomial interpolating the values of a permutation. Given a permutation

$\pi\in\mathfrak S_n$ there is a polynomial

$p$ of minimal degree such that

$p(n)=\pi(n)$ for

$n\in\{1,\dots,n\}$ . This statistic records the degree of

$p$ .

Matching statistic: St000501

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000501: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%

Values

[1] => []
 => []
 => [1] => 1 = 0 + 1
[-1] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[1,2] => []
 => []
 => [1] => 1 = 0 + 1
[1,-2] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[-1,2] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 3 = 2 + 1
[2,1] => []
 => []
 => [1] => 1 = 0 + 1
[2,-1] => [2]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[-2,1] => [2]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[-2,-1] => []
 => []
 => [1] => 1 = 0 + 1
[1,2,3] => []
 => []
 => [1] => 1 = 0 + 1
[1,2,-3] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[1,-2,3] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 3 = 2 + 1
[-1,2,3] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 3 = 2 + 1
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 4 = 3 + 1
[1,3,2] => []
 => []
 => [1] => 1 = 0 + 1
[1,3,-2] => [2]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[1,-3,2] => [2]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[1,-3,-2] => []
 => []
 => [1] => 1 = 0 + 1
[-1,3,2] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 4 = 3 + 1
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 4 = 3 + 1
[-1,-3,-2] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[2,1,3] => []
 => []
 => [1] => 1 = 0 + 1
[2,1,-3] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[2,-1,3] => [2]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 4 = 3 + 1
[-2,1,3] => [2]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 4 = 3 + 1
[-2,-1,3] => []
 => []
 => [1] => 1 = 0 + 1
[-2,-1,-3] => [1]
 => [1,0]
 => [2,1] => 2 = 1 + 1
[2,3,1] => []
 => []
 => [1] => 1 = 0 + 1
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[2,-3,-1] => []
 => []
 => [1] => 1 = 0 + 1
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[-2,3,-1] => []
 => []
 => [1] => 1 = 0 + 1
[-2,-3,1] => []
 => []
 => [1] => 1 = 0 + 1
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[3,1,2] => []
 => []
 => [1] => 1 = 0 + 1
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[3,-1,-2] => []
 => []
 => [1] => 1 = 0 + 1
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[-3,1,-2] => []
 => []
 => [1] => 1 = 0 + 1
[-3,-1,2] => []
 => []
 => [1] => 1 = 0 + 1
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[-6,-4,-2,1,3,5] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[4,5,2,3,-6,1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[2,-6,-4,-5,1,3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[-5,3,6,1,2,4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[4,-5,-6,-3,1,2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[-4,3,-5,2,-6,1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 7 + 1
[5,-3,2,4,8,-7,1,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6 + 1
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7 + 1
[-5,4,-8,-6,-2,1,3,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6 + 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6 + 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 6 + 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7 + 1
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7 + 1
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[-7,-6,3,-8,4,-5,1,2] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6 + 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7 + 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7 + 1
[4,2,-7,8,-6,1,3,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6 + 1
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7 + 1
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => ? = 6 + 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[-7,2,-4,6,-8,-5,1,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6 + 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7 + 1
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 7 + 1
[5,7,-6,4,2,3,-8,1] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 6 + 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => ? = 8 + 1
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,1,2,3,4,7,8,5] => ? = 7 + 1
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 7 + 1
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => ? = 7 + 1
[6,1,2,3,4,-5] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[5,1,2,3,6,-4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[5,1,2,6,4,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[4,1,2,5,6,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[5,1,6,3,4,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[4,1,5,3,6,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[5,1,6,2,3,-4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[3,1,5,6,4,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[4,1,5,6,2,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[3,1,4,5,6,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[5,6,2,3,4,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[4,5,2,3,6,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
[3,5,2,6,4,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1

Description

The size of the first part in the decomposition of a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is defined to be the smallest

$k > 0$ such that

$\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$ . This statistic is undefined for the empty permutation. For the number of parts in the decomposition see [[St000056]].

Matching statistic: St001190
(load all 8 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001190: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%

Values

[1] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-1] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,2] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[1,-2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[-1,2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[-1,-2] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[2,1] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[2,-1] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
[-2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
[-2,-1] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[1,2,3] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[1,2,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,-2,3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[1,-2,-3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[-1,2,3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[-1,2,-3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[-1,-2,3] => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[-1,-2,-3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5 = 3 + 2
[1,3,2] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[1,3,-2] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
[1,-3,2] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
[1,-3,-2] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-1,3,2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[-1,3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 3 + 2
[-1,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 3 + 2
[-1,-3,-2] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[2,1,3] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[2,1,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[2,-1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
[2,-1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 3 + 2
[-2,1,3] => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
[-2,1,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 3 + 2
[-2,-1,3] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-2,-1,-3] => [1]
 => [1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[2,3,1] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[2,3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[2,-3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[2,-3,-1] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[-2,3,-1] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-2,-3,1] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-2,-3,-1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[3,1,2] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[3,1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[3,-1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[3,-1,-2] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[-3,1,-2] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-3,-1,2] => []
 => []
 => [1,0]
 => 2 = 0 + 2
[-3,-1,-2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 3 + 2
[-6,-4,-2,1,3,5] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[4,5,2,3,-6,1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[2,-6,-4,-5,1,3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[-5,3,6,1,2,4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[4,-5,-6,-3,1,2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[-4,3,-5,2,-6,1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[3,2,8,-6,-5,1,4,7] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 2
[5,-3,2,4,8,-7,1,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 2
[4,5,2,3,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[2,-8,-4,-6,-5,1,3,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[2,8,-6,-5,1,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[-5,4,-8,-6,-2,1,3,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 6 + 2
[4,-5,-8,-6,-3,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 6 + 2
[3,-8,-5,-6,-4,1,2,7] => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 6 + 2
[-4,3,-5,2,-8,-6,1,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[5,8,-4,3,-7,1,2,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[-7,-5,4,6,-8,-2,1,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[-4,3,5,8,7,-6,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[-7,-6,3,-8,4,-5,1,2] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 2
[-7,-6,5,3,-8,-4,1,2] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[3,-7,-5,-6,2,4,-8,1] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[4,2,-7,8,-6,1,3,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 2
[-8,-4,-6,2,3,-7,1,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[3,-7,2,8,-5,1,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[-7,6,-8,-4,-2,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 6 + 2
[5,-6,-7,-4,2,3,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[-8,-6,5,-4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[-7,2,-4,6,-8,-5,1,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 2
[-7,6,1,-8,-5,-4,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[-8,6,7,5,-4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[-8,2,5,7,4,-6,1,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 2
[5,7,-6,4,2,3,-8,1] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 6 + 2
[4,-8,-5,-7,1,-6,2,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 8 + 2
[-8,-7,5,6,4,1,2,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 2
[8,6,-7,-4,5,1,2,3] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 2
[5,7,-8,4,3,-6,1,2] => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 2
[6,1,2,3,4,-5] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[5,1,2,3,6,-4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[5,1,2,6,4,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[4,1,2,5,6,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[5,1,6,3,4,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[4,1,5,3,6,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[5,1,6,2,3,-4] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[3,1,5,6,4,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[4,1,5,6,2,-3] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[3,1,4,5,6,-2] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[5,6,2,3,4,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[4,5,2,3,6,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2
[3,5,2,6,4,-1] => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 6 + 2

Description

Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra.

Matching statistic: St000395

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 89%

Values

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

Description

The sum of the heights of the peaks of a Dyck path.

Matching statistic: St000543
(load all 2 compositions to match this statistic)

Mapping

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 89%

Values

[1] => []
 =>  =>  => ? = 0 + 1
[-1] => [1]
 => 10 => 01 => 2 = 1 + 1
[1,2] => []
 =>  =>  => ? = 0 + 1
[1,-2] => [1]
 => 10 => 01 => 2 = 1 + 1
[-1,2] => [1]
 => 10 => 01 => 2 = 1 + 1
[-1,-2] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[2,1] => []
 =>  =>  => ? = 0 + 1
[2,-1] => [2]
 => 100 => 001 => 3 = 2 + 1
[-2,1] => [2]
 => 100 => 001 => 3 = 2 + 1
[-2,-1] => []
 =>  =>  => ? = 0 + 1
[1,2,3] => []
 =>  =>  => ? = 0 + 1
[1,2,-3] => [1]
 => 10 => 01 => 2 = 1 + 1
[1,-2,3] => [1]
 => 10 => 01 => 2 = 1 + 1
[1,-2,-3] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[-1,2,3] => [1]
 => 10 => 01 => 2 = 1 + 1
[-1,2,-3] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[-1,-2,3] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
 => 1110 => 0111 => 4 = 3 + 1
[1,3,2] => []
 =>  =>  => ? = 0 + 1
[1,3,-2] => [2]
 => 100 => 001 => 3 = 2 + 1
[1,-3,2] => [2]
 => 100 => 001 => 3 = 2 + 1
[1,-3,-2] => []
 =>  =>  => ? = 0 + 1
[-1,3,2] => [1]
 => 10 => 01 => 2 = 1 + 1
[-1,3,-2] => [2,1]
 => 1010 => 0011 => 4 = 3 + 1
[-1,-3,2] => [2,1]
 => 1010 => 0011 => 4 = 3 + 1
[-1,-3,-2] => [1]
 => 10 => 01 => 2 = 1 + 1
[2,1,3] => []
 =>  =>  => ? = 0 + 1
[2,1,-3] => [1]
 => 10 => 01 => 2 = 1 + 1
[2,-1,3] => [2]
 => 100 => 001 => 3 = 2 + 1
[2,-1,-3] => [2,1]
 => 1010 => 0011 => 4 = 3 + 1
[-2,1,3] => [2]
 => 100 => 001 => 3 = 2 + 1
[-2,1,-3] => [2,1]
 => 1010 => 0011 => 4 = 3 + 1
[-2,-1,3] => []
 =>  =>  => ? = 0 + 1
[-2,-1,-3] => [1]
 => 10 => 01 => 2 = 1 + 1
[2,3,1] => []
 =>  =>  => ? = 0 + 1
[2,3,-1] => [3]
 => 1000 => 0001 => 4 = 3 + 1
[2,-3,1] => [3]
 => 1000 => 0001 => 4 = 3 + 1
[2,-3,-1] => []
 =>  =>  => ? = 0 + 1
[-2,3,1] => [3]
 => 1000 => 0001 => 4 = 3 + 1
[-2,3,-1] => []
 =>  =>  => ? = 0 + 1
[-2,-3,1] => []
 =>  =>  => ? = 0 + 1
[-2,-3,-1] => [3]
 => 1000 => 0001 => 4 = 3 + 1
[3,1,2] => []
 =>  =>  => ? = 0 + 1
[3,1,-2] => [3]
 => 1000 => 0001 => 4 = 3 + 1
[3,-1,2] => [3]
 => 1000 => 0001 => 4 = 3 + 1
[3,-1,-2] => []
 =>  =>  => ? = 0 + 1
[-3,1,2] => [3]
 => 1000 => 0001 => 4 = 3 + 1
[-3,1,-2] => []
 =>  =>  => ? = 0 + 1
[-3,-1,2] => []
 =>  =>  => ? = 0 + 1
[-3,-1,-2] => [3]
 => 1000 => 0001 => 4 = 3 + 1
[3,2,1] => []
 =>  =>  => ? = 0 + 1
[3,2,-1] => [2]
 => 100 => 001 => 3 = 2 + 1
[3,-2,1] => [1]
 => 10 => 01 => 2 = 1 + 1
[3,-2,-1] => [2,1]
 => 1010 => 0011 => 4 = 3 + 1
[-3,2,1] => [2]
 => 100 => 001 => 3 = 2 + 1
[-3,2,-1] => []
 =>  =>  => ? = 0 + 1
[-3,-2,1] => [2,1]
 => 1010 => 0011 => 4 = 3 + 1
[-3,-2,-1] => [1]
 => 10 => 01 => 2 = 1 + 1
[1,2,3,4] => []
 =>  =>  => ? = 0 + 1
[1,2,3,-4] => [1]
 => 10 => 01 => 2 = 1 + 1
[1,2,-3,4] => [1]
 => 10 => 01 => 2 = 1 + 1
[1,2,-3,-4] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[1,-2,3,4] => [1]
 => 10 => 01 => 2 = 1 + 1
[1,-2,3,-4] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[1,-2,-3,4] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[1,-2,-3,-4] => [1,1,1]
 => 1110 => 0111 => 4 = 3 + 1
[-1,2,3,4] => [1]
 => 10 => 01 => 2 = 1 + 1
[-1,2,3,-4] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[-1,2,-3,4] => [1,1]
 => 110 => 011 => 3 = 2 + 1
[-1,2,-3,-4] => [1,1,1]
 => 1110 => 0111 => 4 = 3 + 1
[1,2,4,3] => []
 =>  =>  => ? = 0 + 1
[1,2,-4,-3] => []
 =>  =>  => ? = 0 + 1
[1,3,2,4] => []
 =>  =>  => ? = 0 + 1
[1,-3,-2,4] => []
 =>  =>  => ? = 0 + 1
[1,3,4,2] => []
 =>  =>  => ? = 0 + 1
[1,3,-4,-2] => []
 =>  =>  => ? = 0 + 1
[1,-3,4,-2] => []
 =>  =>  => ? = 0 + 1
[1,-3,-4,2] => []
 =>  =>  => ? = 0 + 1
[1,4,2,3] => []
 =>  =>  => ? = 0 + 1
[1,4,-2,-3] => []
 =>  =>  => ? = 0 + 1
[1,-4,2,-3] => []
 =>  =>  => ? = 0 + 1
[1,-4,-2,3] => []
 =>  =>  => ? = 0 + 1
[1,4,3,2] => []
 =>  =>  => ? = 0 + 1
[1,-4,3,-2] => []
 =>  =>  => ? = 0 + 1
[2,1,3,4] => []
 =>  =>  => ? = 0 + 1
[-2,-1,3,4] => []
 =>  =>  => ? = 0 + 1
[2,1,4,3] => []
 =>  =>  => ? = 0 + 1
[2,1,-4,-3] => []
 =>  =>  => ? = 0 + 1
[-2,-1,4,3] => []
 =>  =>  => ? = 0 + 1
[-2,-1,-4,-3] => []
 =>  =>  => ? = 0 + 1
[2,3,1,4] => []
 =>  =>  => ? = 0 + 1
[2,-3,-1,4] => []
 =>  =>  => ? = 0 + 1
[-2,3,-1,4] => []
 =>  =>  => ? = 0 + 1
[-2,-3,1,4] => []
 =>  =>  => ? = 0 + 1
[2,3,4,1] => []
 =>  =>  => ? = 0 + 1
[2,3,-4,-1] => []
 =>  =>  => ? = 0 + 1
[2,-3,4,-1] => []
 =>  =>  => ? = 0 + 1
[2,-3,-4,1] => []
 =>  =>  => ? = 0 + 1
[-2,3,4,-1] => []
 =>  =>  => ? = 0 + 1
[-2,3,-4,1] => []
 =>  =>  => ? = 0 + 1

Description

The size of the conjugacy class of a binary word. Two words

$u$ and

$v$ are conjugate, if

$u=w_1 w_2$ and

$v=w_2 w_1$ , see Section 1.3 of [1].

The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000626The minimal period of a binary word. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000144The pyramid weight of the Dyck path. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000924The number of topologically connected components of a perfect matching. St000863The length of the first row of the shifted shape of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001480The number of simple summands of the module J^2/J^3. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000806The semiperimeter of the associated bargraph. St001267The length of the Lyndon factorization of the binary word. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000782The indicator function of whether a given perfect matching is an L & P matching. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.