Identifier
Values
[1] => [1,0] => [1] => 0
[2] => [1,0,1,0] => [1,2] => 0
[1,1] => [1,1,0,0] => [2,1] => 0
[3] => [1,0,1,0,1,0] => [1,2,3] => 0
[2,1] => [1,0,1,1,0,0] => [1,3,2] => 0
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => 0
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 0
[2,2] => [1,1,1,0,0,0] => [3,2,1] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 0
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 0
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 0
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 0
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 0
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 0
[3,3] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 1
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 0
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 0
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 0
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 0
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 0
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 0
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => 0
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => 0
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 2
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => 0
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => 0
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 0
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 2
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => 0
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => 0
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => 0
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8] => 0
[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,8,7] => 0
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => 0
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => 0
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => 3
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 0
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => 4
[4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,3,5,6,7,8,4] => 0
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => 0
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => 3
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,2] => 3
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => 0
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [3,2,4,5,6,7,1] => 0
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,1] => 0
[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8,9] => 0
[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,7,9,8] => 0
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,7,5,6,4] => 0
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => 0
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,6,4,5,7,3] => 4
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => 0
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => 0
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => 0
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => 6
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => 1
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [4,2,3,5,6,7,1] => 0
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,6,4,5,3,2] => 0
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => 0
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [4,3,2,5,6,7,1] => 0
[2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [3,2,4,5,6,7,8,1] => 0
[1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,9,1] => 0
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7,8,9,10] => 0
[9,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,7,8,10,9] => 0
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,7,4,5,6,3] => 0
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => 0
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,7,4,6,5,3] => 0
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => 0
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [5,2,3,4,6,7,1] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 0
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,7,5,6,4,3] => 0
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => 0
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => 0
[3,3,1,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [4,2,3,5,6,7,8,1] => 0
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => 0
[2,2,2,1,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0] => [4,3,2,5,6,7,8,1] => 0
[2,2,1,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,2,4,5,6,7,8,9,1] => 0
[1,1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,9,10,1] => 0
[10,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,6,7,8,9,11,10] => 0
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [6,2,3,4,5,7,1] => 0
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => 0
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => 0
[4,4,1,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0] => [5,2,3,4,6,7,8,1] => 0
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,5,4,3,7,2] => 6
>>> Load all 128 entries. <<<
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [1,7,3,5,6,4,2] => 0
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => 0
[3,3,1,1,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [4,2,3,5,6,7,8,9,1] => 0
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,7,4,5,6,3,2] => 0
[2,2,1,1,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,2,4,5,6,7,8,9,10,1] => 0
[1,1,1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 0
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,7,3,6,5,4,2] => 0
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => 0
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,7,5,4,6,3,2] => 1
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 0
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,7,6,4,5,3,2] => 0
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => 0
[4,3,3,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,8,6,5,4,7,3,2] => 3
[6,5,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [1,8,7,4,5,6,3,2] => 0
[5,4,4,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,8,7,4,6,5,3,2] => 0
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [7,6,5,4,3,2,1] => 0
[4,3,3,3,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,8,7,5,6,4,3,2] => 0
[5,5,4,3] => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [8,2,7,4,6,5,3,1] => 0
[5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,8,7,6,5,4,3,2] => 0
[4,4,4,4,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [7,6,5,4,3,2,8,1] => 0
[] => [] => [] => 0
[5,4,4,4,3] => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [1,9,8,6,5,7,4,3,2] => 1
[4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [8,7,6,5,4,3,2,1] => 0
[5,5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [10,9,8,7,6,5,4,3,2,1] => 0
[5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [9,8,7,6,5,4,3,2,1] => 0
[6,5,5,5] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1,9,8,7,5,6,4,3,2] => 0
[5,4,4,4,4] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,9,8,7,6,5,4,3,2] => 0
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Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..