- FindStat

Your data matches 197 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000237

Mapping

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[(1,2)]
 => [2,1] => [2,1] => 1
[(1,2),(3,4)]
 => [2,1,4,3] => [2,4,1,3] => 1
[(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => 1
[(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,4,6,1,3,5] => 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,6,2,5] => 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,6,1,5] => 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [3,6,2,5,4,1] => 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [4,2,6,5,3,1] => 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [5,4,2,1,6,3] => 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [5,3,1,6,2,4] => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [5,6,2,1,3,4] => 0
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [6,5,2,4,1,3] => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [6,3,5,1,4,2] => 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [4,6,1,5,3,2] => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [5,1,6,4,3,2] => 0
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => 1
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [5,4,3,2,7,1,8,6] => 1
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [4,3,6,2,8,7,5,1] => 2
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [4,3,7,6,2,1,8,5] => 2
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [6,4,3,1,7,2,8,5] => 1
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [4,3,7,1,8,6,5,2] => 1
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [5,3,8,7,6,4,2,1] => 2
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => 0
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,8,6,5,3,7,4,2] => 2
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [6,1,7,5,3,2,8,4] => 2
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,8,4] => 1
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [6,5,3,2,8,7,4,1] => 1
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [5,2,6,3,8,7,4,1] => 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [5,2,7,6,3,1,8,4] => 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [6,5,2,1,7,3,8,4] => 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 1
[(1,6),(2,4),(3,7),(5,8)]
 => [6,4,7,2,8,1,3,5] => [4,2,6,1,7,3,8,5] => 1
[(1,8),(2,4),(3,7),(5,6)]
 => [8,4,7,2,6,5,3,1] => [8,4,7,2,6,5,3,1] => 1
[(1,7),(2,5),(3,8),(4,6)]
 => [7,5,8,6,2,4,1,3] => [5,2,7,1,8,6,4,3] => 0
[(1,8),(2,6),(3,7),(4,5)]
 => [8,6,7,5,4,2,3,1] => [6,2,8,7,5,4,3,1] => 0
[(1,7),(2,6),(3,8),(4,5)]
 => [7,6,8,5,4,2,1,3] => [7,6,2,1,8,5,4,3] => 0
[(1,6),(2,7),(3,8),(4,5)]
 => [6,7,8,5,4,1,2,3] => [6,1,7,2,8,5,4,3] => 0
[(1,3),(2,8),(4,7),(5,6)]
 => [3,8,1,7,6,5,4,2] => [8,7,6,3,5,1,4,2] => 0
[(1,7),(2,8),(3,6),(4,5)]
 => [7,8,6,5,4,3,1,2] => [7,1,8,6,5,4,3,2] => 0
[(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 1

Description

The number of small exceedances. This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Matching statistic: St000502

Mapping

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%

Values

[(1,2)]
 => [2,1] => [2,1] => {{1,2}}
 => 1
[(1,2),(3,4)]
 => [2,1,4,3] => [2,4,1,3] => {{1,2,4},{3}}
 => 1
[(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => {{1,3,4},{2}}
 => 1
[(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
 => 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,4,6,1,3,5] => {{1,2,4},{3,6},{5}}
 => 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
 => 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,6,1,5] => {{1,4,6},{2,3},{5}}
 => 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,2,5,1,6,4] => {{1,3,5,6},{2},{4}}
 => 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [3,6,2,5,4,1] => {{1,3},{2,6},{4,5}}
 => 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [4,2,6,5,3,1] => {{1,4,5},{2},{3,6}}
 => 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [5,4,2,1,6,3] => {{1,5,6},{2,4},{3}}
 => 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => {{1,4},{2},{3,5,6}}
 => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [5,3,1,6,2,4] => {{1,5},{2,3},{4,6}}
 => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [5,6,2,1,3,4] => {{1,5},{2,6},{3},{4}}
 => 0
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [6,5,2,4,1,3] => {{1,6},{2,5},{3},{4}}
 => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [6,3,5,1,4,2] => {{1,6},{2,3,5},{4}}
 => 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [4,6,1,5,3,2] => {{1,4,5},{2,6},{3}}
 => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [5,1,6,4,3,2] => {{1,5},{2},{3,6},{4}}
 => 0
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}}
 => 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => {{1,2,4,8},{3,6},{5},{7}}
 => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => {{1,4},{2},{3,5,7,8},{6}}
 => ? = 1
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [5,4,3,2,7,1,8,6] => {{1,5,7,8},{2,4},{3},{6}}
 => ? = 1
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [4,3,6,2,8,7,5,1] => {{1,4},{2,3,6,7},{5,8}}
 => ? = 2
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [4,3,7,6,2,1,8,5] => {{1,4,6},{2,3,7,8},{5}}
 => ? = 2
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [6,4,3,1,7,2,8,5] => {{1,6},{2,4},{3},{5,7,8}}
 => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [4,3,7,1,8,6,5,2] => {{1,4},{2,3,7},{5,8},{6}}
 => ? = 1
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [5,3,8,7,6,4,2,1] => {{1,5,6},{2,3,8},{4,7}}
 => ? = 2
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => {{1,7},{2,5,8},{3},{4},{6}}
 => ? = 0
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,8,6,5,3,7,4,2] => {{1},{2,8},{3,6,7},{4,5}}
 => ? = 2
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [6,1,7,5,3,2,8,4] => {{1,6},{2},{3,7,8},{4,5}}
 => ? = 2
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,8,4] => {{1,7,8},{2,6},{3,5},{4}}
 => ? = 1
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [6,5,3,2,8,7,4,1] => {{1,6,7},{2,5,8},{3},{4}}
 => ? = 1
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [5,2,6,3,8,7,4,1] => {{1,5,8},{2},{3,6,7},{4}}
 => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [5,2,7,6,3,1,8,4] => {{1,5},{2},{3,7,8},{4,6}}
 => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [6,5,2,1,7,3,8,4] => {{1,6},{2,5,7,8},{3},{4}}
 => ? = 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => {{1,5,7,8},{2},{3,6},{4}}
 => ? = 1
[(1,6),(2,4),(3,7),(5,8)]
 => [6,4,7,2,8,1,3,5] => [4,2,6,1,7,3,8,5] => {{1,4},{2},{3,6},{5,7,8}}
 => ? = 1
[(1,8),(2,4),(3,7),(5,6)]
 => [8,4,7,2,6,5,3,1] => [8,4,7,2,6,5,3,1] => {{1,8},{2,4},{3,7},{5,6}}
 => ? = 1
[(1,7),(2,5),(3,8),(4,6)]
 => [7,5,8,6,2,4,1,3] => [5,2,7,1,8,6,4,3] => {{1,5,8},{2},{3,7},{4},{6}}
 => ? = 0
[(1,8),(2,6),(3,7),(4,5)]
 => [8,6,7,5,4,2,3,1] => [6,2,8,7,5,4,3,1] => {{1,6},{2},{3,8},{4,7},{5}}
 => ? = 0
[(1,7),(2,6),(3,8),(4,5)]
 => [7,6,8,5,4,2,1,3] => [7,6,2,1,8,5,4,3] => {{1,7},{2,6},{3},{4},{5,8}}
 => ? = 0
[(1,6),(2,7),(3,8),(4,5)]
 => [6,7,8,5,4,1,2,3] => [6,1,7,2,8,5,4,3] => {{1,6},{2},{3,7},{4},{5,8}}
 => ? = 0
[(1,3),(2,8),(4,7),(5,6)]
 => [3,8,1,7,6,5,4,2] => [8,7,6,3,5,1,4,2] => {{1,8},{2,7},{3,6},{4},{5}}
 => ? = 0
[(1,7),(2,8),(3,6),(4,5)]
 => [7,8,6,5,4,3,1,2] => [7,1,8,6,5,4,3,2] => {{1,7},{2},{3,8},{4,6},{5}}
 => ? = 0
[(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
 => ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => {{1,2,4,8},{3,6},{5,10},{7},{9}}
 => ? = 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => {{1,10},{2,9},{3,8},{4,7},{5,6}}
 => ? = 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => ?
 => ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => {{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
 => ? = 1

Description

The number of successions of a set partitions. This is the number of indices

$i$ such that

$i$ and

$i+1$ belonging to the same block.

Matching statistic: St000879
(load all 5 compositions to match this statistic)

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000879: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 33%

Values

[(1,2)]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[(1,4),(2,3)]
 => [3,4,2,1] => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 0 = 1 - 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 0 = 1 - 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 0 = 1 - 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 0 = 1 - 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 0 = 1 - 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 0 = 1 - 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ? = 0 - 1
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ? = 0 - 1
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => 0 = 1 - 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [1,4,2,5,3,6] => [1,4,5,2,3,6] => 0 = 1 - 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [1,4,3,6,2,5] => [4,6,3,1,2,5] => ? = 0 - 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [1,4,3,6,2,5] => [4,6,3,1,2,5] => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 1 - 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [1,4,2,5,3,7,6,8] => [1,4,5,7,2,3,6,8] => ? = 1 - 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [1,4,3,7,2,5,6,8] => [7,4,3,1,2,5,6,8] => ? = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [1,4,3,7,5,8,2,6] => [7,4,8,3,5,1,2,6] => ? = 2 - 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? = 2 - 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? = 1 - 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [1,4,3,7,2,6,5,8] => [7,4,3,6,1,2,5,8] => ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? = 2 - 1
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? = 0 - 1
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [1,5,3,7,4,8,2,6] => [5,3,7,8,4,1,2,6] => ? = 2 - 1
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? = 2 - 1
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [1,5,3,7,2,6,4,8] => [7,5,1,3,6,2,4,8] => ? = 1 - 1
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [1,5,3,7,4,8,2,6] => [5,3,7,8,4,1,2,6] => ? = 1 - 1
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? = 1 - 1
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? = 1 - 1
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? = 1 - 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [5,1,2,6,7,3,4,8] => ? = 1 - 1
[(1,6),(2,4),(3,7),(5,8)]
 => [4,6,7,2,8,1,3,5] => [1,4,2,6,3,7,5,8] => [4,1,6,7,2,3,5,8] => ? = 1 - 1
[(1,8),(2,4),(3,7),(5,6)]
 => [4,6,7,2,8,5,3,1] => [1,4,2,6,5,8,3,7] => [6,8,1,4,5,2,3,7] => ? = 1 - 1
[(1,7),(2,5),(3,8),(4,6)]
 => [5,6,7,8,2,4,1,3] => [1,5,2,6,4,8,3,7] => [5,6,8,1,4,2,3,7] => ? = 0 - 1
[(1,8),(2,6),(3,7),(4,5)]
 => [5,6,7,8,4,2,3,1] => [1,5,4,8,2,6,3,7] => [1,8,5,4,6,2,3,7] => ? = 0 - 1
[(1,7),(2,6),(3,8),(4,5)]
 => [5,6,7,8,4,2,1,3] => [1,5,4,8,3,7,2,6] => [8,5,4,7,3,1,2,6] => ? = 0 - 1
[(1,6),(2,7),(3,8),(4,5)]
 => [5,6,7,8,4,1,2,3] => [1,5,4,8,3,7,2,6] => [8,5,4,7,3,1,2,6] => ? = 0 - 1
[(1,3),(2,8),(4,7),(5,6)]
 => [3,6,1,7,8,5,4,2] => [1,3,2,6,5,8,4,7] => [6,8,3,5,1,2,4,7] => ? = 0 - 1
[(1,7),(2,8),(3,6),(4,5)]
 => [5,6,7,8,4,3,1,2] => [1,5,4,8,2,6,3,7] => [1,8,5,4,6,2,3,7] => ? = 0 - 1
[(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => [1,8,5,4,6,2,3,7] => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? = 1 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [1,6,5,10,2,7,4,9,3,8] => [1,10,6,5,7,9,4,2,3,8] => ? = 1 - 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [1,7,6,12,2,8,5,11,3,9,4,10] => [7,12,6,1,2,11,8,5,9,3,4,10] => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1 - 1

Description

The number of long braid edges in the graph of braid moves of a permutation. Given a permutation

$\pi$ , let

$\operatorname{Red}(\pi)$ denote the set of reduced words for

$\pi$ in terms of simple transpositions

$s_i = (i,i+1)$ . We now say that two reduced words are connected by a long braid move if they are obtained from each other by a modification of the form

$s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word. For example, the two reduced words

$s_1s_3s_2s_3$ and

$s_1s_2s_3s_2$ for

$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$ share an edge because they are obtained from each other by interchanging

$s_3s_2s_3 \leftrightarrow s_3s_2s_3$ . This statistic counts the number of such short braid moves among all reduced words.

Matching statistic: St001947
(load all 3 compositions to match this statistic)

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001947: Parking functions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%

Values

[(1,2)]
 => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[(1,2),(3,4)]
 => [2,1,4,3] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[(1,3),(2,4)]
 => [3,4,1,2] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[(1,4),(2,3)]
 => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [4,3,5,2,6,1] => [4,3,5,2,6,1] => 0 = 1 - 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [2,5,3,4,6,1] => [2,5,3,4,6,1] => 0 = 1 - 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [5,4,2,3,6,1] => [5,4,2,3,6,1] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? = 1 - 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,4,2,5,1,3] => [6,4,2,5,1,3] => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [6,1,5,2,3,4] => [6,1,5,2,3,4] => 0 = 1 - 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [5,1,6,2,3,4] => [5,1,6,2,3,4] => 0 = 1 - 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 0 = 1 - 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [4,1,3,6,2,5] => [4,1,3,6,2,5] => ? = 1 - 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [3,2,1,6,4,5] => [3,2,1,6,4,5] => ? = 0 - 1
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [4,3,6,5,1,2] => [4,3,6,5,1,2] => ? = 0 - 1
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [5,6,2,4,1,3] => [5,6,2,4,1,3] => 0 = 1 - 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [2,6,5,1,3,4] => [2,6,5,1,3,4] => 0 = 1 - 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [1,6,5,2,3,4] => [1,6,5,2,3,4] => ? = 0 - 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,1,2,3] => [6,5,4,1,2,3] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,8,1] => ? = 1 - 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [1,3,7,4,5,8,2,6] => [1,3,7,4,5,8,2,6] => ? = 1 - 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [1,7,6,3,4,8,2,5] => [1,7,6,3,4,8,2,5] => ? = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [8,1,6,3,7,2,4,5] => [8,1,6,3,7,2,4,5] => ? = 2 - 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [7,1,6,3,8,2,4,5] => [7,1,6,3,8,2,4,5] => ? = 2 - 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [1,2,7,4,8,3,5,6] => [1,2,7,4,8,3,5,6] => ? = 1 - 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [1,8,6,3,7,2,4,5] => [1,8,6,3,7,2,4,5] => ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [8,7,1,6,2,3,4,5] => [8,7,1,6,2,3,4,5] => ? = 2 - 1
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [1,8,2,7,3,4,5,6] => [1,8,2,7,3,4,5,6] => ? = 0 - 1
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [7,8,1,6,2,3,4,5] => [7,8,1,6,2,3,4,5] => ? = 2 - 1
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [1,7,2,8,3,4,5,6] => [1,7,2,8,3,4,5,6] => ? = 2 - 1
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [7,6,1,8,2,3,4,5] => [7,6,1,8,2,3,4,5] => ? = 1 - 1
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [8,6,1,7,2,3,4,5] => [8,6,1,7,2,3,4,5] => ? = 1 - 1
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [8,1,2,7,3,4,5,6] => [8,1,2,7,3,4,5,6] => ? = 1 - 1
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [7,1,2,8,3,4,5,6] => [7,1,2,8,3,4,5,6] => ? = 1 - 1
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [6,1,2,8,3,4,5,7] => [6,1,2,8,3,4,5,7] => ? = 1 - 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,2,3,8,4,5,6,7] => [1,2,3,8,4,5,6,7] => ? = 1 - 1
[(1,6),(2,4),(3,7),(5,8)]
 => [4,6,7,2,8,1,3,5] => [1,6,2,4,8,3,5,7] => [1,6,2,4,8,3,5,7] => ? = 1 - 1
[(1,8),(2,4),(3,7),(5,6)]
 => [4,6,7,2,8,5,3,1] => [8,6,7,2,5,1,3,4] => [8,6,7,2,5,1,3,4] => ? = 1 - 1
[(1,7),(2,5),(3,8),(4,6)]
 => [5,6,7,8,2,4,1,3] => [1,3,8,7,2,4,5,6] => [1,3,8,7,2,4,5,6] => ? = 0 - 1
[(1,8),(2,6),(3,7),(4,5)]
 => [5,6,7,8,4,2,3,1] => [8,1,7,6,2,3,4,5] => [8,1,7,6,2,3,4,5] => ? = 0 - 1
[(1,7),(2,6),(3,8),(4,5)]
 => [5,6,7,8,4,2,1,3] => [7,1,8,6,2,3,4,5] => [7,1,8,6,2,3,4,5] => ? = 0 - 1
[(1,6),(2,7),(3,8),(4,5)]
 => [5,6,7,8,4,1,2,3] => [1,2,8,7,3,4,5,6] => [1,2,8,7,3,4,5,6] => ? = 0 - 1
[(1,3),(2,8),(4,7),(5,6)]
 => [3,6,1,7,8,5,4,2] => [6,8,3,7,5,1,2,4] => [6,8,3,7,5,1,2,4] => ? = 0 - 1
[(1,7),(2,8),(3,6),(4,5)]
 => [5,6,7,8,4,3,1,2] => [1,8,7,6,2,3,4,5] => [1,8,7,6,2,3,4,5] => ? = 0 - 1
[(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [8,7,6,5,1,2,3,4] => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [6,5,7,4,8,3,9,2,10,1] => ? => ? = 1 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,1,2,3,4,5] => [10,9,8,7,6,1,2,3,4,5] => ? = 1 - 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,1,2,3,4,5,6] => ? => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [7,6,8,5,9,4,10,3,11,2,12,1] => ? => ? = 1 - 1

Description

The number of ties in a parking function. This is the number of indices

$i$ such that

$p_i=p_{i+1}$ .

Matching statistic: St001434
(load all 6 compositions to match this statistic)

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001434: Signed permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%

Values

[(1,2)]
 => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[(1,2),(3,4)]
 => [2,1,4,3] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[(1,3),(2,4)]
 => [3,4,1,2] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[(1,4),(2,3)]
 => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [4,3,5,2,6,1] => [4,3,5,2,6,1] => 0 = 1 - 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [2,5,3,4,6,1] => [2,5,3,4,6,1] => 0 = 1 - 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [5,4,2,3,6,1] => [5,4,2,3,6,1] => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? = 1 - 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,4,2,5,1,3] => [6,4,2,5,1,3] => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [6,1,5,2,3,4] => [6,1,5,2,3,4] => ? = 1 - 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [5,1,6,2,3,4] => [5,1,6,2,3,4] => 0 = 1 - 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 0 = 1 - 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [4,1,3,6,2,5] => [4,1,3,6,2,5] => ? = 1 - 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [3,2,1,6,4,5] => [3,2,1,6,4,5] => ? = 0 - 1
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [4,3,6,5,1,2] => [4,3,6,5,1,2] => ? = 0 - 1
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [5,6,2,4,1,3] => [5,6,2,4,1,3] => 0 = 1 - 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [2,6,5,1,3,4] => [2,6,5,1,3,4] => ? = 1 - 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [1,6,5,2,3,4] => [1,6,5,2,3,4] => ? = 0 - 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,1,2,3] => [6,5,4,1,2,3] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,8,1] => ? = 1 - 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [1,3,7,4,5,8,2,6] => [1,3,7,4,5,8,2,6] => ? = 1 - 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [1,7,6,3,4,8,2,5] => [1,7,6,3,4,8,2,5] => ? = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [8,1,6,3,7,2,4,5] => [8,1,6,3,7,2,4,5] => ? = 2 - 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [7,1,6,3,8,2,4,5] => [7,1,6,3,8,2,4,5] => ? = 2 - 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [1,2,7,4,8,3,5,6] => [1,2,7,4,8,3,5,6] => ? = 1 - 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [1,8,6,3,7,2,4,5] => [1,8,6,3,7,2,4,5] => ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [8,7,1,6,2,3,4,5] => [8,7,1,6,2,3,4,5] => ? = 2 - 1
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [1,8,2,7,3,4,5,6] => [1,8,2,7,3,4,5,6] => ? = 0 - 1
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [7,8,1,6,2,3,4,5] => [7,8,1,6,2,3,4,5] => ? = 2 - 1
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [1,7,2,8,3,4,5,6] => [1,7,2,8,3,4,5,6] => ? = 2 - 1
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [7,6,1,8,2,3,4,5] => [7,6,1,8,2,3,4,5] => ? = 1 - 1
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [8,6,1,7,2,3,4,5] => [8,6,1,7,2,3,4,5] => ? = 1 - 1
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [8,1,2,7,3,4,5,6] => [8,1,2,7,3,4,5,6] => ? = 1 - 1
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [7,1,2,8,3,4,5,6] => [7,1,2,8,3,4,5,6] => ? = 1 - 1
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [6,1,2,8,3,4,5,7] => [6,1,2,8,3,4,5,7] => ? = 1 - 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,2,3,8,4,5,6,7] => [1,2,3,8,4,5,6,7] => ? = 1 - 1
[(1,6),(2,4),(3,7),(5,8)]
 => [4,6,7,2,8,1,3,5] => [1,6,2,4,8,3,5,7] => [1,6,2,4,8,3,5,7] => ? = 1 - 1
[(1,8),(2,4),(3,7),(5,6)]
 => [4,6,7,2,8,5,3,1] => [8,6,7,2,5,1,3,4] => [8,6,7,2,5,1,3,4] => ? = 1 - 1
[(1,7),(2,5),(3,8),(4,6)]
 => [5,6,7,8,2,4,1,3] => [1,3,8,7,2,4,5,6] => [1,3,8,7,2,4,5,6] => ? = 0 - 1
[(1,8),(2,6),(3,7),(4,5)]
 => [5,6,7,8,4,2,3,1] => [8,1,7,6,2,3,4,5] => [8,1,7,6,2,3,4,5] => ? = 0 - 1
[(1,7),(2,6),(3,8),(4,5)]
 => [5,6,7,8,4,2,1,3] => [7,1,8,6,2,3,4,5] => [7,1,8,6,2,3,4,5] => ? = 0 - 1
[(1,6),(2,7),(3,8),(4,5)]
 => [5,6,7,8,4,1,2,3] => [1,2,8,7,3,4,5,6] => [1,2,8,7,3,4,5,6] => ? = 0 - 1
[(1,3),(2,8),(4,7),(5,6)]
 => [3,6,1,7,8,5,4,2] => [6,8,3,7,5,1,2,4] => [6,8,3,7,5,1,2,4] => ? = 0 - 1
[(1,7),(2,8),(3,6),(4,5)]
 => [5,6,7,8,4,3,1,2] => [1,8,7,6,2,3,4,5] => [1,8,7,6,2,3,4,5] => ? = 0 - 1
[(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [8,7,6,5,1,2,3,4] => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [6,5,7,4,8,3,9,2,10,1] => [6,5,7,4,8,3,9,2,10,1] => ? = 1 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,1,2,3,4,5] => [10,9,8,7,6,1,2,3,4,5] => ? = 1 - 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,1,2,3,4,5,6] => [12,11,10,9,8,7,1,2,3,4,5,6] => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [7,6,8,5,9,4,10,3,11,2,12,1] => [7,6,8,5,9,4,10,3,11,2,12,1] => ? = 1 - 1

Description

The number of negative sum pairs of a signed permutation. The number of negative sum pairs of a signed permutation

$\sigma$ is:

$\operatorname{nsp}(\sigma)=\big|\{1\leq i < j\leq n \mid \sigma(i)+\sigma(j) < 0\}\big|,$ see [1, Eq.(8.1)].

Matching statistic: St000882
(load all 3 compositions to match this statistic)

Mapping

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000882: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%

Values

[(1,2)]
 => [2,1] => [1,2] => [1,2] => 1
[(1,2),(3,4)]
 => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[(1,3),(2,4)]
 => [3,4,1,2] => [1,3,2,4] => [3,1,2,4] => 1
[(1,4),(2,3)]
 => [4,3,2,1] => [1,4,2,3] => [3,4,1,2] => 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => [3,4,1,2,5,6] => 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => [3,4,5,1,2,6] => 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => [3,4,5,6,1,2] => 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => [5,3,4,6,1,2] => ? = 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => [5,3,4,1,2,6] => ? = 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => [5,3,1,2,4,6] => ? = 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => [2,5,1,3,4,6] => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ? = 0
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ? = 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => [2,5,6,1,3,4] => 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => [5,2,6,1,3,4] => ? = 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => [5,6,3,1,2,4] => ? = 0
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => [5,6,3,4,1,2] => ? = 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [1,7,2,4,3,5,6,8] => [5,3,4,6,7,1,2,8] => ? = 1
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => [5,6,3,4,7,1,2,8] => ? = 1
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [1,8,2,6,3,4,5,7] => [5,6,7,3,4,8,1,2] => ? = 2
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [1,7,2,6,3,4,5,8] => [5,6,7,3,4,1,2,8] => ? = 2
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [1,6,2,7,3,4,5,8] => [5,6,7,3,1,2,4,8] => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [1,7,2,8,3,4,5,6] => [5,6,7,8,3,1,2,4] => ? = 1
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [7,5,6,8,3,4,1,2] => ? = 2
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => [7,5,6,8,3,1,2,4] => ? = 0
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,6,2,8,3,5,4,7] => [7,5,6,2,8,1,3,4] => ? = 2
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [1,6,2,7,3,5,4,8] => [7,5,6,3,1,2,4,8] => ? = 2
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [7,5,6,3,4,1,2,8] => ? = 1
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [7,5,6,3,4,8,1,2] => ? = 1
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [1,8,2,5,3,6,4,7] => [7,5,3,4,6,8,1,2] => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [7,5,3,4,6,1,2,8] => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [1,6,2,5,3,7,4,8] => [7,5,3,4,1,2,6,8] => ? = 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => [7,5,3,1,2,4,6,8] => ? = 1
[(1,6),(2,4),(3,7),(5,8)]
 => [6,4,7,2,8,1,3,5] => [1,6,2,4,3,7,5,8] => [4,7,3,5,1,2,6,8] => ? = 1
[(1,8),(2,4),(3,7),(5,6)]
 => [8,4,7,2,6,5,3,1] => [1,8,2,4,3,7,5,6] => [4,7,8,3,5,6,1,2] => ? = 1
[(1,7),(2,5),(3,8),(4,6)]
 => [7,5,8,6,2,4,1,3] => [1,7,2,5,3,8,4,6] => [7,4,8,3,5,1,2,6] => ? = 0
[(1,8),(2,6),(3,7),(4,5)]
 => [8,6,7,5,4,2,3,1] => [1,8,2,6,3,7,4,5] => [7,8,5,3,4,6,1,2] => ? = 0
[(1,7),(2,6),(3,8),(4,5)]
 => [7,6,8,5,4,2,1,3] => [1,7,2,6,3,8,4,5] => [7,8,5,3,4,1,2,6] => ? = 0
[(1,6),(2,7),(3,8),(4,5)]
 => [6,7,8,5,4,1,2,3] => [1,6,2,7,3,8,4,5] => [7,8,5,3,1,2,4,6] => ? = 0
[(1,3),(2,8),(4,7),(5,6)]
 => [3,8,1,7,6,5,4,2] => [1,3,2,8,4,7,5,6] => [7,8,2,5,6,1,3,4] => ? = 0
[(1,7),(2,8),(3,6),(4,5)]
 => [7,8,6,5,4,3,1,2] => [1,7,2,8,3,6,4,5] => [7,8,5,6,3,1,2,4] => ? = 0
[(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [7,8,5,6,3,4,1,2] => ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? = 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [1,10,2,9,3,8,4,7,5,6] => ? => ? = 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [12,11,10,9,8,7,6,5,4,3,2,1] => [1,12,2,11,3,10,4,9,5,8,6,7] => ? => ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,3,4,5,6,7,8,9,10,11,12] => [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 1

Description

The number of connected components of short braid edges in the graph of braid moves of a permutation. Given a permutation

$\pi$ , let

$\operatorname{Red}(\pi)$ denote the set of reduced words for

$\pi$ in terms of simple transpositions

$s_i = (i,i+1)$ . We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form

$s_i s_j \leftrightarrow s_j s_i$ for

$|i-j| > 1$ as a consecutive subword of a reduced word. For example, the two reduced words

$s_1s_3s_2$ and

$s_3s_1s_2$ for

$(1243) = (12)(34)(23) = (34)(12)(23)$ share an edge because they are obtained from each other by interchanging

$s_1s_3 \leftrightarrow s_3s_1$ . This statistic counts the number connected components of such short braid moves among all reduced words.

Matching statistic: St001719
(load all 31 compositions to match this statistic)

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%

Values

[(1,2)]
 => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[(1,2),(3,4)]
 => [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[(1,4),(2,3)]
 => [3,4,2,1] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [3,4,2,6,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
 => ? = 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [5,3,6,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [5,3,6,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [4,5,2,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [4,5,2,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
 => ? = 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [5,3,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [5,6,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,9),(5,9),(6,7),(6,10),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
 => ? = 0
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [5,6,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
 => ? = 0
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [3,5,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [4,1,5,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
 => ? = 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [4,5,1,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
 => ? = 0
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [4,5,6,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
 => ? = 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [7,4,2,5,8,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [7,4,5,3,8,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,12),(3,12),(4,12),(5,9),(6,9),(7,10),(8,10),(9,11),(10,12),(11,12)],13)
 => ? = 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [6,7,4,8,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13)
 => ? = 2
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [6,7,4,8,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,10),(5,9),(6,11),(7,11),(8,9),(8,10),(9,12),(10,12),(11,13),(12,13)],14)
 => ? = 2
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [6,7,4,1,8,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [6,7,4,8,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [5,6,7,3,8,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,15),(2,15),(3,15),(4,10),(5,9),(6,11),(7,9),(7,10),(8,11),(8,12),(9,14),(10,14),(11,13),(12,13),(14,15),(15,12)],16)
 => ? = 2
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [5,6,7,3,1,8,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,10),(7,9),(8,9),(8,10),(9,11),(10,11),(11,12)],13)
 => ? = 0
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [5,6,1,7,3,8,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 2
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [5,6,1,7,3,2,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 2
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,15),(2,15),(3,10),(4,9),(5,12),(6,11),(7,11),(7,12),(8,9),(8,10),(9,13),(10,13),(11,14),(12,14),(13,15),(14,15)],16)
 => ? = 1
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [5,6,7,3,2,8,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,10),(4,9),(5,11),(6,11),(7,12),(8,9),(8,10),(9,13),(10,13),(11,14),(13,14),(14,12)],15)
 => ? = 1
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [5,6,2,7,3,8,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12)
 => ? = 1
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [5,6,2,7,3,1,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [5,6,2,1,7,3,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[(1,6),(2,4),(3,7),(5,8)]
 => [4,6,7,2,8,1,3,5] => [6,7,4,2,1,8,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 1
[(1,8),(2,4),(3,7),(5,6)]
 => [4,6,7,2,8,5,3,1] => [6,4,7,8,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12)
 => ? = 1
[(1,7),(2,5),(3,8),(4,6)]
 => [5,6,7,8,2,4,1,3] => [5,6,2,7,1,8,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 0
[(1,8),(2,6),(3,7),(4,5)]
 => [5,6,7,8,4,2,3,1] => [5,6,7,2,8,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,10),(4,9),(5,11),(6,11),(7,12),(8,9),(8,10),(9,13),(10,13),(11,14),(13,14),(14,12)],15)
 => ? = 0
[(1,7),(2,6),(3,8),(4,5)]
 => [5,6,7,8,4,2,1,3] => [5,6,7,2,1,8,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,10),(3,9),(4,12),(5,12),(6,11),(7,11),(8,9),(8,10),(9,13),(10,13),(11,14),(12,14),(13,14)],15)
 => ? = 0
[(1,6),(2,7),(3,8),(4,5)]
 => [5,6,7,8,4,1,2,3] => [5,6,1,7,2,8,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 0
[(1,3),(2,8),(4,7),(5,6)]
 => [3,6,1,7,8,5,4,2] => [6,7,8,3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,13),(6,10),(7,9),(8,9),(8,10),(9,12),(10,12),(12,11),(13,11)],14)
 => ? = 0
[(1,7),(2,8),(3,6),(4,5)]
 => [5,6,7,8,4,3,1,2] => [5,6,7,1,8,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,15),(2,15),(3,10),(4,9),(5,12),(6,11),(7,11),(7,12),(8,9),(8,10),(9,13),(10,13),(11,14),(12,14),(13,15),(14,15)],16)
 => ? = 0
[(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [5,6,7,8,4,3,2,1] => ?
 => ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [6,7,8,9,10,5,4,3,2,1] => ?
 => ? = 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [7,8,9,10,11,12,6,5,4,3,2,1] => ?
 => ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,13),(2,13),(3,13),(4,13),(5,13),(6,13),(7,13),(8,13),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 1

Description

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval

$[a, b]$ in a lattice is small if

$b$ is a join of elements covering

$a$ .

Matching statistic: St000065
(load all 2 compositions to match this statistic)

Mapping

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%

Values

[(1,2)]
 => [2,1] => [2,1] => [[0,1],[1,0]]
 => 0 = 1 - 1
[(1,2),(3,4)]
 => [2,1,4,3] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => 0 = 1 - 1
[(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => 0 = 1 - 1
[(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [1,3,6,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
 => 0 = 1 - 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [1,6,3,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
 => ? = 1 - 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [3,2,1,6,4,5] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,2,5,1,6,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [2,6,3,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 1 - 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [5,4,2,1,6,3] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => 0 = 1 - 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [1,5,6,3,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 1 - 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [5,1,3,6,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 0 - 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [6,5,1,4,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 0 - 1
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [6,1,5,3,4,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
 => ? = 1 - 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [1,6,4,5,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => 0 = 1 - 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [5,1,6,4,3,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 0 - 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [1,3,5,8,2,4,6,7] => [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0]]
 => ? = 1 - 1
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => [[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [5,4,3,2,7,1,8,6] => [[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [4,3,6,2,8,7,5,1] => [[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 2 - 1
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [4,3,7,6,2,1,8,5] => [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 2 - 1
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [6,4,3,1,7,2,8,5] => [[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [4,3,7,1,8,6,5,2] => [[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [5,3,8,7,6,4,2,1] => [[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
 => ? = 2 - 1
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => [[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 0 - 1
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [1,8,6,5,3,7,4,2] => [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0]]
 => ? = 2 - 1
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [6,1,7,5,3,2,8,4] => [[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 2 - 1
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [7,6,5,3,2,1,8,4] => [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [6,5,3,2,8,7,4,1] => [[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 1 - 1
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [5,2,6,3,8,7,4,1] => [[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 1 - 1
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [5,2,7,6,3,1,8,4] => [[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,6),(2,5),(3,7),(4,8)]
 => [6,5,7,8,2,1,3,4] => [6,5,2,1,7,3,8,4] => [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,6),(2,4),(3,7),(5,8)]
 => [6,4,7,2,8,1,3,5] => [4,2,6,1,7,3,8,5] => [[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0]]
 => ? = 1 - 1
[(1,8),(2,4),(3,7),(5,6)]
 => [8,4,7,2,6,5,3,1] => [8,2,7,4,6,5,3,1] => [[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0]]
 => ? = 1 - 1
[(1,7),(2,5),(3,8),(4,6)]
 => [7,5,8,6,2,4,1,3] => [5,2,7,1,8,6,4,3] => [[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 0 - 1
[(1,8),(2,6),(3,7),(4,5)]
 => [8,6,7,5,4,2,3,1] => [6,2,8,7,5,4,3,1] => [[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0]]
 => ? = 0 - 1
[(1,7),(2,6),(3,8),(4,5)]
 => [7,6,8,5,4,2,1,3] => [7,6,2,1,8,5,4,3] => [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 0 - 1
[(1,6),(2,7),(3,8),(4,5)]
 => [6,7,8,5,4,1,2,3] => [6,1,7,2,8,5,4,3] => [[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0]]
 => ? = 0 - 1
[(1,3),(2,8),(4,7),(5,6)]
 => [3,8,1,7,6,5,4,2] => [8,7,6,1,5,3,4,2] => [[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0]]
 => ? = 0 - 1
[(1,7),(2,8),(3,6),(4,5)]
 => [7,8,6,5,4,3,1,2] => [7,1,8,6,5,4,3,2] => [[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0]]
 => ? = 0 - 1
[(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0]]
 => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [1,3,5,7,10,2,4,6,8,9] => ?
 => ? = 1 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => [[0,0,0,0,0,0,0,0,0,1],[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],[0,0,0,0,1,0,0,0,0,0],[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 - 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => [[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0]]
 => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,3,5,7,9,12,2,4,6,8,10,11] => ?
 => ? = 1 - 1

Description

The number of entries equal to -1 in an alternating sign matrix. The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.

Matching statistic: St001720
(load all 31 compositions to match this statistic)

Mapping

Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%

Values

[(1,2)]
 => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[(1,2),(3,4)]
 => [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[(1,4),(2,3)]
 => [3,4,2,1] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2 = 1 + 1
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [3,4,2,6,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
 => ? = 1 + 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [5,3,6,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [5,3,6,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [4,5,2,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1 + 1
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [4,5,2,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
 => ? = 1 + 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [5,3,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [5,6,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,9),(5,9),(6,7),(6,10),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
 => ? = 0 + 1
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [5,6,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
 => ? = 0 + 1
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [3,5,6,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [4,1,5,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
 => ? = 1 + 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [4,5,1,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
 => ? = 0 + 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [4,5,6,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
 => ? = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [7,4,2,5,8,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [7,4,5,3,8,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,12),(3,12),(4,12),(5,9),(6,9),(7,10),(8,10),(9,11),(10,12),(11,12)],13)
 => ? = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [6,7,4,8,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,10),(5,10),(6,9),(7,9),(8,11),(9,12),(10,12),(12,11)],13)
 => ? = 2 + 1
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [6,7,4,8,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,10),(5,9),(6,11),(7,11),(8,9),(8,10),(9,12),(10,12),(11,13),(12,13)],14)
 => ? = 2 + 1
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [6,7,4,1,8,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 1 + 1
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [6,7,4,8,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 1 + 1
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [5,6,7,3,8,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,15),(2,15),(3,15),(4,10),(5,9),(6,11),(7,9),(7,10),(8,11),(8,12),(9,14),(10,14),(11,13),(12,13),(14,15),(15,12)],16)
 => ? = 2 + 1
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [5,6,7,3,1,8,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,10),(7,9),(8,9),(8,10),(9,11),(10,11),(11,12)],13)
 => ? = 0 + 1
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [5,6,1,7,3,8,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 2 + 1
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [5,6,1,7,3,2,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 2 + 1
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,15),(2,15),(3,10),(4,9),(5,12),(6,11),(7,11),(7,12),(8,9),(8,10),(9,13),(10,13),(11,14),(12,14),(13,15),(14,15)],16)
 => ? = 1 + 1
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [5,6,7,3,2,8,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,10),(4,9),(5,11),(6,11),(7,12),(8,9),(8,10),(9,13),(10,13),(11,14),(13,14),(14,12)],15)
 => ? = 1 + 1
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [5,6,2,7,3,8,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12)
 => ? = 1 + 1
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [5,6,2,7,3,1,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 1 + 1
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [5,6,2,1,7,3,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 1 + 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[(1,6),(2,4),(3,7),(5,8)]
 => [4,6,7,2,8,1,3,5] => [6,7,4,2,1,8,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 1 + 1
[(1,8),(2,4),(3,7),(5,6)]
 => [4,6,7,2,8,5,3,1] => [6,4,7,8,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,11),(6,9),(7,9),(8,10),(9,11),(11,10)],12)
 => ? = 1 + 1
[(1,7),(2,5),(3,8),(4,6)]
 => [5,6,7,8,2,4,1,3] => [5,6,2,7,1,8,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 0 + 1
[(1,8),(2,6),(3,7),(4,5)]
 => [5,6,7,8,4,2,3,1] => [5,6,7,2,8,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,10),(4,9),(5,11),(6,11),(7,12),(8,9),(8,10),(9,13),(10,13),(11,14),(13,14),(14,12)],15)
 => ? = 0 + 1
[(1,7),(2,6),(3,8),(4,5)]
 => [5,6,7,8,4,2,1,3] => [5,6,7,2,1,8,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,10),(3,9),(4,12),(5,12),(6,11),(7,11),(8,9),(8,10),(9,13),(10,13),(11,14),(12,14),(13,14)],15)
 => ? = 0 + 1
[(1,6),(2,7),(3,8),(4,5)]
 => [5,6,7,8,4,1,2,3] => [5,6,1,7,2,8,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(2,11),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,11),(10,11)],12)
 => ? = 0 + 1
[(1,3),(2,8),(4,7),(5,6)]
 => [3,6,1,7,8,5,4,2] => [6,7,8,3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,13),(6,10),(7,9),(8,9),(8,10),(9,12),(10,12),(12,11),(13,11)],14)
 => ? = 0 + 1
[(1,7),(2,8),(3,6),(4,5)]
 => [5,6,7,8,4,3,1,2] => [5,6,7,1,8,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,15),(2,15),(3,10),(4,9),(5,12),(6,11),(7,11),(7,12),(8,9),(8,10),(9,13),(10,13),(11,14),(12,14),(13,15),(14,15)],16)
 => ? = 0 + 1
[(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [5,6,7,8,4,3,2,1] => ?
 => ? = 1 + 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,4,6,8,10,1,3,5,7,9] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [6,7,8,9,10,5,4,3,2,1] => ?
 => ? = 1 + 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [7,8,9,10,11,12,6,5,4,3,2,1] => [7,8,9,10,11,12,6,5,4,3,2,1] => ?
 => ? = 1 + 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,13),(2,13),(3,13),(4,13),(5,13),(6,13),(7,13),(8,13),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 1 + 1

Description

The minimal length of a chain of small intervals in a lattice. An interval

$[a, b]$ is small if

$b$ is a join of elements covering

$a$ .

Matching statistic: St000928

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000928: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%

Values

[(1,2)]
 => [1,0]
 => []
 => ? = 1 - 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1]
 => ? = 1 - 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => []
 => ? = 1 - 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => []
 => ? = 1 - 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [2,1]
 => 0 = 1 - 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => 0 = 1 - 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => 0 = 1 - 1
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1]
 => ? = 1 - 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1]
 => ? = 1 - 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => []
 => ? = 1 - 1
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => []
 => ? = 1 - 1
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => []
 => ? = 1 - 1
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1]
 => ? = 1 - 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1]
 => -1 = 0 - 1
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1]
 => -1 = 0 - 1
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1]
 => ? = 1 - 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => []
 => ? = 1 - 1
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => []
 => ? = 0 - 1
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => []
 => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0 = 1 - 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 = 1 - 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 2 - 1
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 2 - 1
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 1 - 1
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 - 1
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 0 - 1
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 - 1
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 - 1
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,6),(2,4),(3,7),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 1 - 1
[(1,8),(2,4),(3,7),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 1 - 1
[(1,7),(2,5),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 0 - 1
[(1,8),(2,6),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 0 - 1
[(1,7),(2,6),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 0 - 1
[(1,6),(2,7),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 0 - 1
[(1,3),(2,8),(4,7),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => -1 = 0 - 1
[(1,7),(2,8),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 0 - 1
[(1,8),(2,7),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 0 = 1 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => ? = 1 - 1

Description

The sum of the coefficients of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1].

The following 187 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001820The size of the image of the pop stack sorting operator. St001568The smallest positive integer that does not appear twice in the partition. St001846The number of elements which do not have a complement in the lattice. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St000069The number of maximal elements of a poset. St001260The permanent of an alternating sign matrix. St000894The trace of an alternating sign matrix. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000881The number of short braid edges in the graph of braid moves of a permutation. St000068The number of minimal elements in a poset. St000741The Colin de Verdière graph invariant. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000893The number of distinct diagonal sums of an alternating sign matrix. St001964The interval resolution global dimension of a poset. St000958The number of Bruhat factorizations of a permutation. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St001948The number of augmented double ascents of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000907The number of maximal antichains of minimal length in a poset. St000943The number of spots the most unlucky car had to go further in a parking function. St001429The number of negative entries in a signed permutation. St001520The number of strict 3-descents. St001555The order of a signed permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St000037The sign of a permutation. St000056The decomposition (or block) number of a permutation. St000100The number of linear extensions of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000633The size of the automorphism group of a poset. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001344The neighbouring number of a permutation. St001399The distinguishing number of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001590The crossing number of a perfect matching. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001851The number of Hecke atoms of a signed permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001864The number of excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000102The charge of a semistandard tableau. St000221The number of strong fixed points of a permutation. St000239The number of small weak excedances. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000308The height of the tree associated to a permutation. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000709The number of occurrences of 14-2-3 or 14-3-2. St000750The number of occurrences of the pattern 4213 in a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000850The number of 1/2-balanced pairs in a poset. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000962The 3-shifted major index of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

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

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001430The number of positive entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001488The number of corners of a skew partition. St001524The degree of symmetry of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001545The second Elser number of a connected graph. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001684The reduced word complexity of a permutation. St001715The number of non-records in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001741The largest integer such that all patterns of this size are contained in the permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001845The number of join irreducibles minus the rank of a lattice. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000401The size of the symmetry class of a permutation. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St001817The number of flag weak exceedances of a signed permutation.