Your data matches 274 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000183
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1] => [1]
 => 1
[2]
 => [1,0,1,0]
 => [1,2] => [1,1]
 => 1
[1,1]
 => [1,1,0,0]
 => [2,1] => [2]
 => 1
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 1
[2,2]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,1,1]
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [2,1,1,1]
 => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1]
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
 => 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,1,1,1]
 => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [2,2,1]
 => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,1,1,1]
 => 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 2
[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] => [2,1,1,1,1,1]
 => 1
[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] => [1,1,1,1,1,1,1,1]
 => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => [2,1,1,1,1]
 => 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1]
 => 1
[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] => [1,1,1,1,1,1,1,1,1]
 => 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[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] => [1,1,1,1,1,1,1,1,1,1]
 => 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,1,2] => [2,1,1,1,1]
 => 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,2,3,4,5] => [2,1,1,1,1]
 => 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,1,6,2,3,4] => [2,2,1,1]
 => 2
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,1,2,3,4] => [2,1,1,1,1]
 => 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,2,3,4,5,6] => [2,1,1,1,1,1]
 => 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
 => 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
 => 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
 => 1
Description
The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.
Matching statistic: St000920
(load all 3 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 2
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 1
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Matching statistic: St000994
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [2,1] => 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,3,1] => 1
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,7,9,1,8] => 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => 2
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,8,1,3,4,5,6,7] => 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,1,2,3,4,5,6,7] => 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [8,1,2,3,4,5,6,9,7] => 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$. Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000862
(load all 11 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0]
 => [1] => 1
[2]
 => [1,0,1,0]
 => [2,1] => 1
[1,1]
 => [1,1,0,0]
 => [1,2] => 1
[3]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[2,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,1,3] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,1,7] => 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1] => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,2,1,8] => 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3,2,1] => 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [10,9,8,7,6,5,4,3,2,1] => 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [3,1,2,4,5,6] => 2
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => 1
[]
 => []
 => [] => ? = 0
Description
The number of parts of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of parts of the shifted shape.
Matching statistic: St000068
(load all 4 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0]
 => [.,.]
 => ([],1)
 => 1
[2]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1
[1,1]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1
[3]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1
[2,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2]
 => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 2
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [[[[[[.,[.,.]],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],.],.],.],.],.],.]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[]
 => []
 => .
 => ?
 => ? = 0
Description
The number of minimal elements in a poset.
Matching statistic: St000071
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0]
 => [.,.]
 => ([],1)
 => 1
[2]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1
[1,1]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1
[3]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1
[2,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2]
 => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 2
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [[[[[[.,[.,.]],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],.],.],.],.],.],.]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[]
 => []
 => .
 => ?
 => ? = 0
Description
The number of maximal chains in a poset.
Matching statistic: St000527
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0]
 => [.,.]
 => ([],1)
 => 1
[2]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1
[1,1]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => 1
[3]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1
[2,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2]
 => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 2
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [[[[[[.,[.,.]],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[[[[[.,.],.],.],.],.],.],.]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => 1
[]
 => []
 => .
 => ?
 => ? = 0
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St001175
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0]
 => {{1}}
 => [1]
 => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => {{1},{2}}
 => [1,1]
 => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => {{1,2}}
 => [2]
 => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => [1,1,1]
 => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => [2,1]
 => 0 = 1 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => [2,1]
 => 0 = 1 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => [1,1,1,1]
 => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => [2,1,1]
 => 0 = 1 - 1
[2,2]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => [3]
 => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => [2,1,1]
 => 0 = 1 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => [2,1,1]
 => 0 = 1 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => 0 = 1 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => [3,1]
 => 0 = 1 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => 0 = 1 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => [2,2]
 => 1 = 2 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => 0 = 1 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => 0 = 1 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => [3,1,1]
 => 0 = 1 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => [3,1]
 => 0 = 1 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => [2,2,1]
 => 1 = 2 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => [4]
 => 0 = 1 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => [2,2,1]
 => 1 = 2 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1},{2,6},{3},{4},{5}}
 => [2,1,1,1,1]
 => 0 = 1 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => [2,1,1,1,1]
 => 0 = 1 - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1]
 => 0 = 1 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => [3,1,1]
 => 0 = 1 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => [2,2,1]
 => 1 = 2 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => [4,1]
 => 0 = 1 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => [3,2]
 => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,7},{2},{3},{4},{5},{6}}
 => [2,1,1,1,1,1]
 => 0 = 1 - 1
[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}}
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => [3,1,1]
 => 0 = 1 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4,5,6}}
 => [4,1,1]
 => 0 = 1 - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => [4,1]
 => 0 = 1 - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => [4,1]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => [2,1,1,1,1,1,1]
 => 0 = 1 - 1
[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}}
 => [1,1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => [5]
 => 0 = 1 - 1
[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}}
 => [1,1,1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => {{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => 0 = 1 - 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => [5,1]
 => 0 = 1 - 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => {{1,2,3,6},{4,5}}
 => [4,2]
 => 1 = 2 - 1
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => {{1,2,3,4,6},{5}}
 => [5,1]
 => 0 = 1 - 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => [6]
 => 0 = 1 - 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,7}}
 => [6,1]
 => 0 = 1 - 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,7},{6}}
 => [6,1]
 => 0 = 1 - 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => {{1,2,3,4,5,6,7}}
 => [7]
 => 0 = 1 - 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,7,8}}
 => [7,1]
 => 0 = 1 - 1
[]
 => []
 => {}
 => ?
 => ? = 0 - 1
Description
The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001673
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0]
 => [1] => [1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [2,1] => [2] => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [1,2] => [2] => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [2,3,1] => [3] => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => [3] => 0 = 1 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [3,1,2] => [3] => 0 = 1 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4] => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [4] => 0 = 1 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => [3] => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4] => 0 = 1 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4] => 0 = 1 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5] => 0 = 1 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [5] => 0 = 1 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [4] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5] => 0 = 1 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,3] => 1 = 2 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5] => 0 = 1 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5] => 0 = 1 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,1] => [6] => 0 = 1 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [5] => 0 = 1 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4] => 0 = 1 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,3] => 1 = 2 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4] => 0 = 1 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [1,4] => 1 = 2 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,1,3] => [6] => 0 = 1 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,1,2] => [6] => 0 = 1 - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,1] => [7] => 0 = 1 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5] => 0 = 1 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [2,3] => 1 = 2 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [5] => 0 = 1 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [2,3] => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,1,2] => [7] => 0 = 1 - 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => [8] => 0 = 1 - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => [5] => 0 = 1 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,5,6] => [6] => 0 = 1 - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => [5] => 0 = 1 - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => [5] => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,8,1,2] => [8] => 0 = 1 - 1
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,1] => [9] => 0 = 1 - 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [5] => 0 = 1 - 1
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,10,1] => [10] => 0 = 1 - 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,5,6,1,2,3] => [6] => 0 = 1 - 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [6] => 0 = 1 - 1
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6,2,3,4,5] => [1,5] => 1 = 2 - 1
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,1,2,3,4,5] => [6] => 0 = 1 - 1
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => [6] => 0 = 1 - 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => [7] => 0 = 1 - 1
[3,3,3,3,3]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => [7] => 0 = 1 - 1
[4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [7] => 0 = 1 - 1
[5,4,4,4]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => [8] => 0 = 1 - 1
[]
 => []
 => [] => ? => ? = 0 - 1
Description
The degree of asymmetry of an integer composition. This is the number of pairs of symmetrically positioned distinct entries.
Matching statistic: St000157
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [[1],[2]]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [[1,2],[3]]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[1,2,3],[4]]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [[1,2,3,4],[5]]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [[1,2,4],[3]]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [[1,3,4],[2]]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,3,4,5],[2]]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [[1,2,3,4,5],[6]]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [[1,2,3,5],[4]]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [[1,2],[3,4]]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [[1,3,4,5],[2]]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [[1,3,4,5,6],[2]]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7]]
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [[1,2,3],[4,5]]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [[1,2,3],[4,5]]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[1,2,5],[3,4]]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[1,3,4],[2,5]]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => [[1,3,4,5,6],[2]]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [[1,3,4,5,6,7],[2]]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7],[8]]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [[1,2,3],[4,5]]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [[1,2,4],[3,5]]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [[1,2,5],[3,4]]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [[1,3,5],[2,4]]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [[1,3,4,5,6,7,8],[2]]
 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [[1,2,3,4,5,6,7,8],[9]]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [[1,2,3,4],[5,6]]
 => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [[1,2,5],[3,4]]
 => 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => [[1,2,5,6],[3,4]]
 => 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [[1,3,4,5,6,7,8,9],[2]]
 => 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9],[10]]
 => 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [4,5,6,1,2,3] => [[1,2,3],[4,5,6]]
 => 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 1
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => [[1,2,3,4,5],[6,7]]
 => 1
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [4,5,1,6,2,3] => [[1,2,3],[4,5,6]]
 => 1
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,4,6,1,2,5] => [[1,2,5],[3,4,6]]
 => 2
[4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [5,6,7,1,2,3,4] => [[1,2,3,4],[5,6,7]]
 => 1
[3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => [[1,2,3,7],[4,5,6]]
 => 1
[4,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [4,5,6,1,7,2,3] => [[1,2,3,7],[4,5,6]]
 => 1
[3,3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,1,2,3] => [[1,2,3,7,8],[4,5,6]]
 => 1
[4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,1,2,3,4] => [[1,2,3,4],[5,6,7,8]]
 => 1
[5,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
 => [5,6,7,1,8,2,3,4] => [[1,2,3,4],[5,6,7,8]]
 => 1
[]
 => []
 => [] => []
 => 0
[5,5,5,5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,7,8,9,10,11,1,2,3,4,5] => ?
 => ? = 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
The following 264 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000291The number of descents of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001394The genus of a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000820The number of compositions obtained by rotating the composition. St000903The number of different parts of an integer composition. St000408The number of occurrences of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St000761The number of ascents in an integer composition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000386The number of factors DDU in a Dyck path. St001712The number of natural descents of a standard Young tableau. St000201The number of leaf nodes in a binary tree. St000679The pruning number of an ordered tree. St000232The number of crossings of a set partition. St001584The area statistic between a Dyck path and its bounce path. St000396The register function (or Horton-Strahler number) of a binary tree. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000124The cardinality of the preimage of the Simion-Schmidt map. St000568The hook number of a binary tree. St001732The number of peaks visible from the left. St000496The rcs statistic of a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000491The number of inversions of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000298The order dimension or Dushnik-Miller dimension of a poset. St000482The (zero)-forcing number of a graph. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000632The jump number of the poset. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001307The number of induced stars on four vertices in a graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001638The book thickness of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000358The number of occurrences of the pattern 31-2. St001727The number of invisible inversions of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000919The number of maximal left branches of a binary tree. St000996The number of exclusive left-to-right maxima of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St001549The number of restricted non-inversions between exceedances. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000886The number of permutations with the same antidiagonal sums. St001735The number of permutations with the same set of runs. St000142The number of even parts of a partition. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000663The number of right floats of a permutation. St000710The number of big deficiencies of a permutation. St001092The number of distinct even parts of a partition. St001513The number of nested exceedences of a permutation. St001728The number of invisible descents of a permutation. St000092The number of outer peaks of a permutation. St000252The number of nodes of degree 3 of a binary tree. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000099The number of valleys of a permutation, including the boundary. St000354The number of recoils of a permutation. St001235The global 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. 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. St000397The Strahler number of a rooted tree. St000523The number of 2-protected nodes of a rooted tree. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001578The minimal number of edges to add or remove to make a graph a line graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001964The interval resolution global dimension of a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000570The Edelman-Greene number of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001220The width of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001570The minimal number of edges to add to make a graph Hamiltonian. St000021The number of descents of a permutation. St001729The number of visible descents of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. 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 $eA$. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000864The number of circled entries of the shifted recording tableau of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the 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 $K[x]/(x^n)$. St001960The number of descents of a permutation minus one if its first entry is not one. St000732The number of double deficiencies of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000367The number of simsun double descents of a permutation. St000664The number of right ropes of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000709The number of occurrences of 14-2-3 or 14-3-2. St001330The hat guessing number of a graph. St001868The number of alignments of type NE of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001621The number of atoms of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001896The number of right descents of a signed permutations. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001867The number of alignments of type EN of a signed permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed 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. St000417The size of the automorphism group of the ordered tree. St000943The number of spots the most unlucky car had to go further in a parking function. St001058The breadth of the ordered tree. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001410The minimal entry of a semistandard tableau. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000264The girth of a graph, which is not a tree. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001720The minimal length of a chain of small intervals in a lattice. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000039The number of crossings of a permutation. St000084The number of subtrees. St000091The descent variation of a composition. St000105The number of blocks in the set partition. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000328The maximum number of child nodes in a tree. St000355The number of occurrences of the pattern 21-3. St000365The number of double ascents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000462The major index minus the number of excedences of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000823The number of unsplittable factors of the set partition. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001537The number of cyclic crossings of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000879The number of long braid edges in the graph of braid moves of a permutation. St001624The breadth of a lattice. St001413Half the length of the longest even length palindromic prefix of a binary word. St001569The maximal modular displacement of a permutation.