searching the database
Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000476
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> 0
[2,-1] => [2]
=> [1,0,1,0]
=> 1
[-2,1] => [2]
=> [1,0,1,0]
=> 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> 0
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> 0
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 2
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 2
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 2
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 2
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 2
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 2
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 2
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 2
[3,2,-1] => [2]
=> [1,0,1,0]
=> 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 0
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> 0
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> 0
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 0
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> 1
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> 0
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> 0
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 1
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000840
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000840: Perfect matchings ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000840: Perfect matchings ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[2,-1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[-2,1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[3,2,-1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 2
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> ? = 5
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 6
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 6
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4
[-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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 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]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4
[-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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 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]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 6
[-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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 6
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> ? = 4
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 6
[-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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 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]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> ? = 5
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 6
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> ? = 4
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> ? = 5
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> ? = 5
[3,4,5,6,7,1,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[3,4,6,1,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[4,1,5,6,7,3,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[4,1,6,3,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[5,6,2,7,4,1,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[6,1,2,-7,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[-7,4,-3,1,2,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> ? = 5
[6,1,7,-5,2,3,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[7,-5,-4,-6,1,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[-2,-5,-7,1,3,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[7,-6,2,1,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[-3,-5,7,-6,1,2,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[5,4,1,-7,2,3,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[7,4,1,-6,2,3,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[-5,4,1,-7,-6,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[3,1,6,2,4,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 7
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 7
[4,6,1,2,3,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[6,1,5,2,4,8,7,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[6,5,1,2,3,8,7,-4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 7
[1,7,5,3,8,4,6,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 6
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 7
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 7
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 7
[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]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 7
Description
The number of closers smaller than the largest opener in a perfect matching.
An opener (or left hand endpoint) of a perfect matching is a number that is matched with a larger number, which is then called a closer (or right hand endpoint).
Matching statistic: St000199
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[2,-1] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[-2,1] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-6,-4,-2,1,3,5] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,5,2,3,-6,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[2,-6,-4,-5,1,3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[-5,3,6,1,2,4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,-5,-6,-3,1,2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[-4,3,-5,2,-6,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 + 1
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 4 + 1
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 3 + 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 3 + 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 1
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 4 + 1
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 1
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 + 1
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 + 1
[6,1,2,3,4,-5] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,1,2,3,6,-4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,1,2,6,4,-3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,1,2,5,6,-3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,1,6,3,4,-2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,1,5,3,6,-2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,1,6,2,3,-4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[3,1,5,6,4,-2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,1,5,6,2,-3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[3,1,4,5,6,-2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,6,2,3,4,-1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,5,2,3,6,-1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[3,5,2,6,4,-1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
Description
The column of the unique '1' in the last row of the alternating sign matrix.
Matching statistic: St000200
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000200: Alternating sign matrices ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000200: Alternating sign matrices ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[2,-1] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[-2,1] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[-6,-4,-2,1,3,5] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,5,2,3,-6,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[2,-6,-4,-5,1,3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[-5,3,6,1,2,4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,-5,-6,-3,1,2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[-4,3,-5,2,-6,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 + 1
[5,-3,2,4,8,-7,1,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4 + 1
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 1
[-7,-6,3,-8,4,-5,1,2] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4 + 1
[4,2,-7,8,-6,1,3,5] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 1
[-7,2,-4,6,-8,-5,1,3] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 + 1
[5,7,-6,4,2,3,-8,1] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 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]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 6 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 4 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 + 1
[6,1,2,3,4,-5] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,1,2,3,6,-4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,1,2,6,4,-3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,1,2,5,6,-3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,1,6,3,4,-2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,1,5,3,6,-2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,1,6,2,3,-4] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[3,1,5,6,4,-2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,1,5,6,2,-3] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[3,1,4,5,6,-2] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[5,6,2,3,4,-1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[4,5,2,3,6,-1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[3,5,2,6,4,-1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
Description
The row of the unique '1' in the last column of the alternating sign matrix.
Matching statistic: St000740
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[-2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 1 + 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 1 + 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 1 + 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 1 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 1 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3 = 2 + 1
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 3 = 2 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 3 = 2 + 1
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 3 + 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] => ? = 6 + 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] => ? = 6 + 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] => ? = 4 + 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] => ? = 3 + 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] => ? = 3 + 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] => ? = 3 + 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] => ? = 6 + 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] => ? = 4 + 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] => ? = 4 + 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] => ? = 6 + 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] => ? = 3 + 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] => ? = 4 + 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] => ? = 4 + 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] => ? = 3 + 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] => ? = 4 + 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] => ? = 6 + 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] => ? = 6 + 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] => ? = 4 + 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] => ? = 6 + 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] => ? = 6 + 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] => ? = 3 + 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] => ? = 6 + 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] => ? = 4 + 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] => ? = 5 + 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] => ? = 3 + 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] => ? = 6 + 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] => ? = 4 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 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] => ? = 5 + 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001291
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,-3] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,-4,3] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,-2,4] => [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[-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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[-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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[-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]
=> ? = 5 + 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]
=> ? = 5 + 1
[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]
=> ? = 3 + 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]
=> ? = 6 + 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]
=> ? = 6 + 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]
=> ? = 4 + 1
[-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]
=> ? = 3 + 1
[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]
=> ? = 3 + 1
[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]
=> ? = 3 + 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]
=> ? = 6 + 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]
=> ? = 4 + 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]
=> ? = 4 + 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]
=> ? = 6 + 1
[-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]
=> ? = 3 + 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]
=> ? = 4 + 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]
=> ? = 4 + 1
[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]
=> ? = 3 + 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]
=> ? = 4 + 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]
=> ? = 6 + 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]
=> ? = 6 + 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]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 4 + 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]
=> ? = 6 + 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]
=> ? = 6 + 1
[-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]
=> ? = 3 + 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]
=> ? = 6 + 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]
=> ? = 4 + 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]
=> ? = 5 + 1
[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]
=> ? = 3 + 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]
=> ? = 6 + 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]
=> ? = 4 + 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]
=> ? = 5 + 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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
[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]
=> ? = 5 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!