- FindStat

Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000337
(load all 5 compositions to match this statistic)

Mapping

Mp00058: Perfect matchings —to permutation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[(1,2)]
 => [2,1] => 1
[(1,2),(3,4)]
 => [2,1,4,3] => 2
[(1,3),(2,4)]
 => [3,4,1,2] => 2
[(1,4),(2,3)]
 => [4,3,2,1] => 2
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 3
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => 3
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 3
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => 3
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => 2
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => 3
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => 2
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => 3
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => 3
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => 3
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 3
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => 3
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => 2
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => 3
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => 4
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => 4
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => 4
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => 3
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => 4
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => 3
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => 4
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => 5
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => 4
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => 4
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => 4
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => 4
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => 4
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => 4
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => 4
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => 4
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => 4
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => 3
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => 4
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => 4
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => 4
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => 4
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => 4
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => 4
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => 3
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => 3
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => 3

Description

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$ where

$\textrm{inv} \, \tau_{i}$ is the number of inversions of

$\tau_{i}$ .

Matching statistic: St001907
(load all 2 compositions to match this statistic)

Mapping

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001907: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%

Values

[(1,2)]
 => [2,1] => [2,1] => [2,1] => 1
[(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[(1,3),(2,4)]
 => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2
[(1,4),(2,3)]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [2,4,1,3,6,5] => [2,4,1,3,6,5] => 3
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 3
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [4,6,1,5,3,2] => [4,6,1,5,3,2] => ? = 3
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [5,4,1,6,3,2] => [5,4,1,6,3,2] => ? = 2
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [3,1,5,2,6,4] => [3,1,5,2,6,4] => 3
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [3,6,2,1,5,4] => [3,6,2,1,5,4] => ? = 2
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [2,3,6,1,4,5] => [2,3,6,1,4,5] => 3
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [4,6,2,5,1,3] => [4,6,2,5,1,3] => 3
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,4,6,3,5] => [2,1,4,6,3,5] => 3
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 3
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [5,4,2,6,1,3] => [5,4,2,6,1,3] => 3
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [3,2,6,5,1,4] => [3,2,6,5,1,4] => ? = 2
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [2,6,4,3,1,5] => [2,6,4,3,1,5] => ? = 3
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [2,4,1,3,6,5,8,7] => [2,4,1,3,6,5,8,7] => ? = 4
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 4
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [4,6,1,5,3,2,8,7] => [4,6,1,5,3,2,8,7] => ? = 4
[(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => [5,4,1,6,3,2,8,7] => [5,4,1,6,3,2,8,7] => ? = 3
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [6,8,1,5,4,7,3,2] => [6,8,1,5,4,7,3,2] => ? = 4
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [7,6,1,5,4,8,3,2] => [7,6,1,5,4,8,3,2] => ? = 3
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [7,6,1,3,5,2,8,4] => [7,6,1,3,5,2,8,4] => ? = 4
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [6,8,1,3,5,2,7,4] => [6,8,1,3,5,2,7,4] => ? = 5
[(1,6),(2,4),(3,5),(7,8)]
 => [6,4,5,2,3,1,8,7] => [3,1,5,2,6,4,8,7] => [3,1,5,2,6,4,8,7] => ? = 4
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [3,6,2,1,5,4,8,7] => [3,6,2,1,5,4,8,7] => ? = 3
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [2,3,6,1,4,5,8,7] => [2,3,6,1,4,5,8,7] => ? = 4
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [4,6,2,5,1,3,8,7] => [4,6,2,5,1,3,8,7] => ? = 4
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [2,1,4,6,3,5,8,7] => [2,1,4,6,3,5,8,7] => ? = 4
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => 4
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [5,4,2,6,1,3,8,7] => [5,4,2,6,1,3,8,7] => ? = 4
[(1,4),(2,6),(3,5),(7,8)]
 => [4,6,5,1,3,2,8,7] => [3,2,6,5,1,4,8,7] => [3,2,6,5,1,4,8,7] => ? = 3
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [2,6,4,3,1,5,8,7] => [2,6,4,3,1,5,8,7] => ? = 4
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 4
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [6,8,1,7,5,4,3,2] => [6,8,1,7,5,4,3,2] => ? = 4
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [7,6,1,8,5,4,3,2] => [7,6,1,8,5,4,3,2] => ? = 3
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [5,1,7,2,8,6,4,3] => [5,1,7,2,8,6,4,3] => ? = 4
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [5,8,2,1,7,6,4,3] => [5,8,2,1,7,6,4,3] => ? = 3
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [2,5,8,1,7,4,3,6] => [2,5,8,1,7,4,3,6] => ? = 4
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [6,8,2,7,4,3,1,5] => [6,8,2,7,4,3,1,5] => ? = 4
[(1,4),(2,7),(3,5),(6,8)]
 => [4,7,5,1,3,8,2,6] => [6,8,2,3,7,5,1,4] => [6,8,2,3,7,5,1,4] => ? = 4
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [6,8,2,5,4,7,1,3] => [6,8,2,5,4,7,1,3] => ? = 4
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [2,1,6,8,3,7,5,4] => [2,1,6,8,3,7,5,4] => ? = 4
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => [2,1,7,6,3,8,5,4] => [2,1,7,6,3,8,5,4] => ? = 3
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [7,6,2,5,4,8,1,3] => [7,6,2,5,4,8,1,3] => ? = 3
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [7,6,2,3,8,5,1,4] => [7,6,2,3,8,5,1,4] => ? = 3
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [7,6,2,8,4,3,1,5] => [7,6,2,8,4,3,1,5] => ? = 3
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [5,2,7,1,8,4,3,6] => [5,2,7,1,8,4,3,6] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [2,6,5,1,8,4,3,7] => [2,6,5,1,8,4,3,7] => ? = 3
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [6,5,2,1,8,7,4,3] => [6,5,2,1,8,7,4,3] => ? = 4
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [4,2,1,6,3,8,7,5] => [4,2,1,6,3,8,7,5] => ? = 4
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [2,4,1,6,3,8,5,7] => [2,4,1,6,3,8,5,7] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [4,2,7,3,1,8,5,6] => [4,2,7,3,1,8,5,6] => ? = 4
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [4,2,3,7,1,5,8,6] => [4,2,3,7,1,5,8,6] => ? = 4
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [5,2,7,3,8,6,1,4] => [5,2,7,3,8,6,1,4] => ? = 3
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [5,2,7,4,6,8,1,3] => [5,2,7,4,6,8,1,3] => ? = 4
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [2,1,5,3,7,4,8,6] => [2,1,5,3,7,4,8,6] => ? = 4
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [2,1,5,8,4,3,7,6] => [2,1,5,8,4,3,7,6] => ? = 3
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [5,8,4,2,6,7,1,3] => [5,8,4,2,6,7,1,3] => ? = 3
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [5,8,3,2,7,6,1,4] => [5,8,3,2,7,6,1,4] => ? = 2
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [3,2,4,8,1,5,7,6] => [3,2,4,8,1,5,7,6] => ? = 3
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [2,4,8,3,1,7,5,6] => [2,4,8,3,1,7,5,6] => ? = 3
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [4,8,3,2,1,7,6,5] => [4,8,3,2,1,7,6,5] => ? = 3
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [4,1,7,3,2,8,6,5] => [4,1,7,3,2,8,6,5] => ? = 3
[(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => [2,1,4,3,8,7,6,5] => 4
[(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => 4
[(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => 4
[(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 4

Description

The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. For a signed permutation

$\sigma$ , this equals

$\left\lfloor \dfrac{fexc(\sigma)+1}{2} \right\rfloor = exc(\sigma) + \left\lfloor \dfrac{neg(\sigma)+1}{2} \right\rfloor,$ where

$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$

$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|,$

$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|.$ This statistic has the same distribution as the descent statistic [[St001427]].