searching the database
Your data matches 242 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [3] => [3]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> 1
Description
The length of the partition.
Matching statistic: St000147
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [2]
=> 2
[1,1,0,0]
=> [2] => [2]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [3]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [2,1]
=> 2
[1,1,0,1,0,0]
=> [3] => [3]
=> [1,1,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [3,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [3,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [3,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [3,2]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2,2,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [3,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St000378
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [2]
=> 2
[1,1,0,0]
=> [2] => [2]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [2,1]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [3]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [3]
=> 2
[1,1,0,1,0,0]
=> [3] => [3]
=> [1,1,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [3,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [2,2]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [2,2]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [2,2]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [4]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [3,2]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [4,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [5]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [5]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [4,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [5]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000228
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 100%●distinct values known / distinct values provided: 91%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 100%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [1,1] => [2] => [2]
=> 2
[1,1,0,0]
=> [2] => [1] => [1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [3]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,1]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,1]
=> 2
[1,1,0,1,0,0]
=> [3] => [1] => [1]
=> 1
[1,1,1,0,0,0]
=> [3] => [1] => [1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [2,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [2,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [3,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [2,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [2,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [2,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [2,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [3,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [2,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => [11]
=> ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1,1,1,1] => [1,1,9] => [9,1,1]
=> ? = 11
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1,1] => [1,10] => [10,1]
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,2,1] => [9,1,1] => ?
=> ? = 11
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000395
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 99%●distinct values known / distinct values provided: 73%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 99%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [2] => [1] => [1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[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] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[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] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => ?
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,2,1,1] => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,2,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,3,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1,1] => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,2,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1,1] => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1,1] => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,2,1,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,3,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1,1] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1,1] => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 9
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,3,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,1,3,1] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,2,1] => [8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,3,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,3,1] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,3,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,1,1,1,1,1,2,1] => [1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1,1,1,1] => [1,1,9] => ?
=> ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,1,1,1,2] => [1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 9
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1,1] => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,2,1] => [9,1,1] => ?
=> ? = 11
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001020
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 99%●distinct values known / distinct values provided: 64%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 99%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [2] => [1] => [1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[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] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[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] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => ?
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,2,1,1] => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,2,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,3,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1,1] => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,2,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1,1] => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1,1] => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,2,1,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,3,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1,1] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1,1] => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,1,1,2] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,1,1,1,1,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 8
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,2,1,1,1,1,1] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1,1,1] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,1,1,2] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,1,1,1,1,2,1] => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,1,1,2,1,1] => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 8
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,2,1,1,1,1,1] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,3,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,1,3,1] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000998
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 99%●distinct values known / distinct values provided: 64%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 99%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [2] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 + 1
[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] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[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] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => ?
=> ? = 11 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,2,1,1] => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,2,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,3,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1,1] => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,2,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1,1] => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1,1] => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,2,1,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,3,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1,1] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1,1] => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,1,1,2] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,1,1,1,1,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,2,1,1,1,1,1] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1,1,1] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,1,1,2] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,1,1,1,1,2,1] => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 8 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,1,1,2,1,1] => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 8 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,2,1,1,1,1,1] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,3,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,1,3,1] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9 + 1
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000144
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 97%●distinct values known / distinct values provided: 55%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 97%●distinct values known / distinct values provided: 55%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [2] => [1] => [1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7
[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] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[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] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => ?
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,1,2] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,2,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St001018
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 97%●distinct values known / distinct values provided: 55%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 97%●distinct values known / distinct values provided: 55%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [2] => [1] => [1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7
[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] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[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] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => ?
=> ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 7
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,1,2] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,2,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001240
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001240: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 97%●distinct values known / distinct values provided: 55%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001240: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 97%●distinct values known / distinct values provided: 55%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [2] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 + 1
[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] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[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] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [11] => ?
=> ? = 11 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1,1] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 8 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,1,1,1,1,2] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,2,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 + 1
Description
The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
The following 232 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St000288The number of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000011The number of touch points (or returns) of a Dyck path. St000382The first part of an integer composition. St000306The bounce count of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000297The number of leading ones in a binary word. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000678The number of up steps after the last double rise of a Dyck path. St000383The last part of an integer composition. St000439The position of the first down step of a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000675The number of centered multitunnels of a Dyck path. St000617The number of global maxima of a Dyck path. St000759The smallest missing part in an integer partition. St000054The first entry of the permutation. St000925The number of topologically connected components of a set partition. St000098The chromatic number of a graph. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000068The number of minimal elements in a poset. St001050The number of terminal closers of a set partition. St000237The number of small exceedances. St000475The number of parts equal to 1 in a partition. St000504The cardinality of the first block of a set partition. St000031The number of cycles in the cycle decomposition of a permutation. St000971The smallest closer of a set partition. St000069The number of maximal elements of a poset. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000502The number of successions of a set partitions. St000286The number of connected components of the complement of a graph. St001733The number of weak left to right maxima of a Dyck path. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000544The cop number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001777The number of weak descents in an integer composition. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000273The domination number of a graph. St000822The Hadwiger number of the graph. St000916The packing number of a graph. St001479The number of bridges of a graph. St001812The biclique partition number of a graph. St001316The domatic number of a graph. St001829The common independence number of a graph. St001644The dimension of a graph. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000052The number of valleys of a Dyck path not on the x-axis. St000214The number of adjacencies of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000676The number of odd rises of a Dyck path. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000553The number of blocks of a graph. St000843The decomposition number of a perfect matching. St001461The number of topologically connected components of the chord diagram of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000993The multiplicity of the largest part of an integer partition. St000740The last entry of a permutation. St000990The first ascent of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000648The number of 2-excedences of a permutation. St000738The first entry in the last row of a standard tableau. St000203The number of external nodes of a binary tree. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000734The last entry in the first row of a standard tableau. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000883The number of longest increasing subsequences of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000087The number of induced subgraphs. St000352The Elizalde-Pak rank of a permutation. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000456The monochromatic index of a connected graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St001110The 3-dynamic chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001828The Euler characteristic of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St000061The number of nodes on the left branch of a binary tree. St001480The number of simple summands of the module J^2/J^3. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000160The multiplicity of the smallest part of a partition. St001545The second Elser number of a connected graph. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000022The number of fixed points of a permutation. St000924The number of topologically connected components of a perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000742The number of big ascents of a permutation after prepending zero. St000241The number of cyclical small excedances. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 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 $eA$. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001552The number of inversions between excedances and fixed points of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000386The number of factors DDU in a Dyck path. St001889The size of the connectivity set of a signed permutation. St001862The number of crossings of a signed permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000649The number of 3-excedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001557The number of inversions of the second entry of a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001621The number of atoms of a lattice.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!