Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000319
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => [1]
 => 0
[1,1] => [1,1,0,0]
 => [2,1] => [2]
 => 1
[1,2] => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[2,1] => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[1,1,1] => [1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => 1
[1,1,2] => [1,1,0,1,0,0]
 => [2,3,1] => [3]
 => 2
[1,2,1] => [1,1,0,1,0,0]
 => [2,3,1] => [3]
 => 2
[2,1,1] => [1,1,0,1,0,0]
 => [2,3,1] => [3]
 => 2
[1,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,3,1] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[3,1,1] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[2,1,2] => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[2,2,1] => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,3,2] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[2,1,3] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[2,3,1] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[3,1,2] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[3,2,1] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => 3
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => 3
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => 3
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => 3
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => [1]
 => 0
[1,1] => [1,1,0,0]
 => [2,1] => [2]
 => 1
[1,2] => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[2,1] => [1,0,1,0]
 => [1,2] => [1,1]
 => 0
[1,1,1] => [1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => 1
[1,1,2] => [1,1,0,1,0,0]
 => [2,3,1] => [3]
 => 2
[1,2,1] => [1,1,0,1,0,0]
 => [2,3,1] => [3]
 => 2
[2,1,1] => [1,1,0,1,0,0]
 => [2,3,1] => [3]
 => 2
[1,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,3,1] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[3,1,1] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[2,1,2] => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[2,2,1] => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,3,2] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[2,1,3] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[2,3,1] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[3,1,2] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[3,2,1] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => 3
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => 3
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => 3
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => 3
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 3
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001232
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 0 + 1
[2,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 0 + 1
[1,1,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 1 + 1
[1,1,2] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,2,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[2,1,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[1,3,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[3,1,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[1,2,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 1 + 1
[2,1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 1 + 1
[2,2,1] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[1,3,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[2,1,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[2,3,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[3,1,2] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[3,2,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2 + 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2 + 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2 + 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2 + 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2 + 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2 + 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[3,2,1,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? = 1 + 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? = 1 + 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? = 1 + 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? = 1 + 1
[1,1,2,3,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,2,4,3] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,3,2,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,3,4,2] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,4,2,3] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,4,3,2] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,1,3,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,1,4,3] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,3,1,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,3,4,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,4,1,3] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,4,3,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,1,2,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,1,4,2] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,2,1,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,2,4,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,4,1,2] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,4,2,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,1,2,3] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,1,3,2] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,2,1,3] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,2,3,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,3,1,2] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,3,2,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,1,1,3,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,1,1,4,3] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,1,3,1,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,1,3,4,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,1,4,1,3] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,1,4,3,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,3,1,1,4] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,3,1,4,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,3,4,1,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000317
Mapping
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
 => [(1,2)]
 => [2,1] => 0
[1,1] => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1
[1,2] => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0
[2,1] => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 0
[1,1,1] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 1
[1,1,2] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2
[1,2,1] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2
[2,1,1] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2
[1,1,3] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => 1
[1,3,1] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => 1
[3,1,1] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => 1
[1,2,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1
[2,1,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1
[2,2,1] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[1,3,2] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[2,1,3] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[2,3,1] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[3,1,2] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[3,2,1] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 0
[1,1,1,1] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? = 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 3
[1,1,2,1] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 3
[1,2,1,1] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 3
[2,1,1,1] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 3
[1,1,1,3] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? = 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? = 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? = 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? = 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => ? = 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[3,2,1,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 3
[1,1,2,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[2,4,1,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[4,1,1,2] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[4,1,2,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[4,2,1,1] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? = 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => ? = 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => ? = 1
Description
The cycle descent number of a permutation. Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.