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Your data matches 433 different statistics following compositions of up to 3 maps.
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Matching statistic: St000377
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1]
=> [3]
=> 1 = 0 + 1
[3] => [3]
=> [1,1,1]
=> 2 = 1 + 1
[1,1,2] => [2,1,1]
=> [2,2]
=> 1 = 0 + 1
[1,2,1] => [2,1,1]
=> [2,2]
=> 1 = 0 + 1
[1,3] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[2,2] => [2,2]
=> [4]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[4] => [4]
=> [1,1,1,1]
=> 3 = 2 + 1
[1,1,1,2] => [2,1,1,1]
=> [3,1,1]
=> 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
=> [3,1,1]
=> 1 = 0 + 1
[1,1,3] => [3,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,1,1] => [2,1,1,1]
=> [3,1,1]
=> 1 = 0 + 1
[1,2,2] => [2,2,1]
=> [2,2,1]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,4] => [4,1]
=> [2,1,1,1]
=> 3 = 2 + 1
[2,1,2] => [2,2,1]
=> [2,2,1]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [2,2,1]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [5]
=> 3 = 2 + 1
[3,1,1] => [3,1,1]
=> [4,1]
=> 2 = 1 + 1
[3,2] => [3,2]
=> [5]
=> 3 = 2 + 1
[4,1] => [4,1]
=> [2,1,1,1]
=> 3 = 2 + 1
[5] => [5]
=> [1,1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [4,2]
=> 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [4,2]
=> 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
=> [3,3]
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [4,2]
=> 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[1,1,3,1] => [3,1,1,1]
=> [3,3]
=> 2 = 1 + 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 3 = 2 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [4,2]
=> 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[1,2,3] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[1,3,1,1] => [3,1,1,1]
=> [3,3]
=> 2 = 1 + 1
[1,3,2] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 3 = 2 + 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 4 = 3 + 1
[2,1,1,2] => [2,2,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[2,1,3] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[2,2,1,1] => [2,2,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[2,2,2] => [2,2,2]
=> [2,2,2]
=> 3 = 2 + 1
[2,3,1] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[2,4] => [4,2]
=> [2,2,1,1]
=> 4 = 3 + 1
[3,1,1,1] => [3,1,1,1]
=> [3,3]
=> 2 = 1 + 1
[3,1,2] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[3,2,1] => [3,2,1]
=> [5,1]
=> 3 = 2 + 1
[3,3] => [3,3]
=> [6]
=> 4 = 3 + 1
[4,1,1] => [4,1,1]
=> [3,1,1,1]
=> 3 = 2 + 1
[4,2] => [4,2]
=> [2,2,1,1]
=> 4 = 3 + 1
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[3] => [3]
=> [1,1,1]
=> 2 = 1 + 1
[1,1,2] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[1,2,1] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[1,3] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[2,2] => [2,2]
=> [2,2]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[4] => [4]
=> [1,1,1,1]
=> 3 = 2 + 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 1 = 0 + 1
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 1 = 0 + 1
[1,2,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[1,4] => [4,1]
=> [2,1,1,1]
=> 3 = 2 + 1
[2,1,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [2,2,1]
=> 3 = 2 + 1
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[3,2] => [3,2]
=> [2,2,1]
=> 3 = 2 + 1
[4,1] => [4,1]
=> [2,1,1,1]
=> 3 = 2 + 1
[5] => [5]
=> [1,1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 3 = 2 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,2,3] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 3 = 2 + 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 4 = 3 + 1
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[2,1,3] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[2,2,1,1] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[2,2,2] => [2,2,2]
=> [3,3]
=> 3 = 2 + 1
[2,3,1] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[2,4] => [4,2]
=> [2,2,1,1]
=> 4 = 3 + 1
[3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> 2 = 1 + 1
[3,1,2] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[3,2,1] => [3,2,1]
=> [3,2,1]
=> 3 = 2 + 1
[3,3] => [3,3]
=> [2,2,2]
=> 4 = 3 + 1
[4,1,1] => [4,1,1]
=> [3,1,1,1]
=> 3 = 2 + 1
[4,2] => [4,2]
=> [2,2,1,1]
=> 4 = 3 + 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000010
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ 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
[1,2] => [2,1] => [2,1]
=> 2 = 0 + 2
[3] => [1,1,1] => [1,1,1]
=> 3 = 1 + 2
[1,1,2] => [3,1] => [3,1]
=> 2 = 0 + 2
[1,2,1] => [2,2] => [2,2]
=> 2 = 0 + 2
[1,3] => [2,1,1] => [2,1,1]
=> 3 = 1 + 2
[2,2] => [1,2,1] => [2,1,1]
=> 3 = 1 + 2
[3,1] => [1,1,2] => [2,1,1]
=> 3 = 1 + 2
[4] => [1,1,1,1] => [1,1,1,1]
=> 4 = 2 + 2
[1,1,1,2] => [4,1] => [4,1]
=> 2 = 0 + 2
[1,1,2,1] => [3,2] => [3,2]
=> 2 = 0 + 2
[1,1,3] => [3,1,1] => [3,1,1]
=> 3 = 1 + 2
[1,2,1,1] => [2,3] => [3,2]
=> 2 = 0 + 2
[1,2,2] => [2,2,1] => [2,2,1]
=> 3 = 1 + 2
[1,3,1] => [2,1,2] => [2,2,1]
=> 3 = 1 + 2
[1,4] => [2,1,1,1] => [2,1,1,1]
=> 4 = 2 + 2
[2,1,2] => [1,3,1] => [3,1,1]
=> 3 = 1 + 2
[2,2,1] => [1,2,2] => [2,2,1]
=> 3 = 1 + 2
[2,3] => [1,2,1,1] => [2,1,1,1]
=> 4 = 2 + 2
[3,1,1] => [1,1,3] => [3,1,1]
=> 3 = 1 + 2
[3,2] => [1,1,2,1] => [2,1,1,1]
=> 4 = 2 + 2
[4,1] => [1,1,1,2] => [2,1,1,1]
=> 4 = 2 + 2
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> 5 = 3 + 2
[1,1,1,1,2] => [5,1] => [5,1]
=> 2 = 0 + 2
[1,1,1,2,1] => [4,2] => [4,2]
=> 2 = 0 + 2
[1,1,1,3] => [4,1,1] => [4,1,1]
=> 3 = 1 + 2
[1,1,2,1,1] => [3,3] => [3,3]
=> 2 = 0 + 2
[1,1,2,2] => [3,2,1] => [3,2,1]
=> 3 = 1 + 2
[1,1,3,1] => [3,1,2] => [3,2,1]
=> 3 = 1 + 2
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> 4 = 2 + 2
[1,2,1,1,1] => [2,4] => [4,2]
=> 2 = 0 + 2
[1,2,1,2] => [2,3,1] => [3,2,1]
=> 3 = 1 + 2
[1,2,2,1] => [2,2,2] => [2,2,2]
=> 3 = 1 + 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> 4 = 2 + 2
[1,3,1,1] => [2,1,3] => [3,2,1]
=> 3 = 1 + 2
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> 4 = 2 + 2
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> 4 = 2 + 2
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> 5 = 3 + 2
[2,1,1,2] => [1,4,1] => [4,1,1]
=> 3 = 1 + 2
[2,1,2,1] => [1,3,2] => [3,2,1]
=> 3 = 1 + 2
[2,1,3] => [1,3,1,1] => [3,1,1,1]
=> 4 = 2 + 2
[2,2,1,1] => [1,2,3] => [3,2,1]
=> 3 = 1 + 2
[2,2,2] => [1,2,2,1] => [2,2,1,1]
=> 4 = 2 + 2
[2,3,1] => [1,2,1,2] => [2,2,1,1]
=> 4 = 2 + 2
[2,4] => [1,2,1,1,1] => [2,1,1,1,1]
=> 5 = 3 + 2
[3,1,1,1] => [1,1,4] => [4,1,1]
=> 3 = 1 + 2
[3,1,2] => [1,1,3,1] => [3,1,1,1]
=> 4 = 2 + 2
[3,2,1] => [1,1,2,2] => [2,2,1,1]
=> 4 = 2 + 2
[3,3] => [1,1,2,1,1] => [2,1,1,1,1]
=> 5 = 3 + 2
[4,1,1] => [1,1,1,3] => [3,1,1,1]
=> 4 = 2 + 2
[4,2] => [1,1,1,2,1] => [2,1,1,1,1]
=> 5 = 3 + 2
Description
The length of the partition.
Matching statistic: St000225
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 3
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 3
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 3
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 3
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St001124
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 3
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 3
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 3
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 3
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000147
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2 = 1 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 2 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 2 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 2 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 3 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 4 = 3 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 3 = 2 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> 3 = 2 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 4 = 3 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> 3 = 2 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 4 = 3 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 3 = 2 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 4 = 3 + 1
Description
The largest part of an integer partition.
Matching statistic: St000228
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3] => [3]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[1,2,1] => [2,1,1]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 1 + 1
[2,2] => [2,2]
=> [2,2]
=> [2]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 1 + 1
[4] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 2 + 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,2,2] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 2 + 1
[2,1,2] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [2,2,1]
=> [2,1]
=> 3 = 2 + 1
[3,1,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 1 + 1
[3,2] => [3,2]
=> [2,2,1]
=> [2,1]
=> 3 = 2 + 1
[4,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 2 + 1
[5] => [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2 = 1 + 1
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 3 = 2 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2 = 1 + 1
[1,2,3] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 3 = 2 + 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 4 = 3 + 1
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2 = 1 + 1
[2,1,3] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[2,2,1,1] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2 = 1 + 1
[2,2,2] => [2,2,2]
=> [3,3]
=> [3]
=> 3 = 2 + 1
[2,3,1] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[2,4] => [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 4 = 3 + 1
[3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2 = 1 + 1
[3,1,2] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[3,2,1] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 3 = 2 + 1
[3,3] => [3,3]
=> [2,2,2]
=> [2,2]
=> 4 = 3 + 1
[4,1,1] => [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 3 = 2 + 1
[4,2] => [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 4 = 3 + 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000369
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3] => [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,2] => [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,1] => [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,2] => [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,1,2] => [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,1,1] => [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,2] => [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,2] => [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,1] => [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2] => [3,2]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 2 + 1
[4,1] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[5] => [5]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,3,1] => [3,1,1,1]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [3,2,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,1,1] => [3,1,1,1]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [3,2,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[2,1,1,2] => [2,2,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [3,2,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[2,2,1,1] => [2,2,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,2] => [2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,1] => [3,2,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[2,4] => [4,2]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[3,1,1,1] => [3,1,1,1]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [3,2,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [3,2,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[3,3] => [3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 4 = 3 + 1
[4,1,1] => [4,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[4,2] => [4,2]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
Description
The dinv deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$
In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$.
See also [[St000376]] for the bounce deficit.
Matching statistic: St000394
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4 = 3 + 1
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,1,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,3,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[3,1,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[4,1,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,2] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000533
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2 = 1 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 2 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 2 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 2 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 3 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 4 = 3 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 3 = 2 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1]
=> 3 = 2 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 4 = 3 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> 3 = 2 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 4 = 3 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 3 = 2 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 4 = 3 + 1
Description
The minimum of the number of parts and the size of the first part of an integer partition.
This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
The following 423 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000783The side length of the largest staircase partition fitting into a partition. St000144The pyramid weight of the Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000378The diagonal inversion number of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000759The smallest missing part in an integer partition. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra 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. St000507The number of ascents of a standard tableau. St000074The number of special entries. St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000288The number of ones in a binary word. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000480The number of lower covers of a partition in dominance order. St000159The number of distinct parts of the integer partition. St001432The order dimension of the partition. St000691The number of changes of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001486The number of corners of the ribbon associated with an integer composition. St000013The height of a Dyck path. St001484The number of singletons of an integer partition. 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. St000011The number of touch points (or returns) of a Dyck path. St000097The order of the largest clique of the graph. St000012The area of a Dyck path. St000984The number of boxes below precisely one peak. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001581The achromatic number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000442The maximal area to the right of an up step of a Dyck path. St000481The number of upper covers of a partition in dominance order. St000439The position of the first down step of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001058The breadth of the ordered tree. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000306The bounce count of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000931The number of occurrences of the pattern UUU in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000098The chromatic number of a graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000445The number of rises of length 1 of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000617The number of global maxima of a Dyck path. St000703The number of deficiencies of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000141The maximum drop size of a permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St000675The number of centered multitunnels of a Dyck path. St000167The number of leaves of an ordered tree. St000925The number of topologically connected components of a set partition. St001050The number of terminal closers of a set partition. St000366The number of double descents of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000734The last entry in the first row of a standard tableau. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000874The position of the last double rise in a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000521The number of distinct subtrees of an ordered tree. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000502The number of successions of a set partitions. St000536The pathwidth of a graph. St000728The dimension of a set partition. St000778The metric dimension 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. 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. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000025The number of initial rises of a Dyck path. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000636The hull number of a graph. St000808The number of up steps of the associated bargraph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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:
St001342The number of vertices in the center of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000632The jump number of the poset. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001479The number of bridges of a graph. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001498The normalised height of a Nakayama algebra with magnitude 1. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000105The number of blocks in the set partition. St000153The number of adjacent cycles of a permutation. St000273The domination number of a graph. St000383The last part of an integer composition. St000504The cardinality of the first block of a set partition. St000527The width of the poset. St000544The cop number of a graph. St000668The least common multiple of the parts of the partition. St000686The finitistic dominant dimension of a Dyck path. St000708The product of the parts of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000822The Hadwiger number of the graph. St000916The packing number of a graph. St000971The smallest closer of a set partition. St001062The maximal size of a block of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001279The sum of the parts of an integer partition that are at least two. 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. St001389The number of partitions of the same length below the given integer partition. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001812The biclique partition number of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000308The height of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St001315The dissociation number of a graph. St000019The cardinality of the support of a permutation. St000214The number of adjacencies of a permutation. St000018The number of inversions of a permutation. St000058The order of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000731The number of double exceedences of a permutation. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St000237The number of small exceedances. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000246The number of non-inversions of a permutation. St000883The number of longest increasing subsequences of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000446The disorder of a permutation. St000911The number of maximal antichains of maximal size in a poset. St000702The number of weak deficiencies of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000292The number of ascents of a binary word. St000539The number of odd inversions of a permutation. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000989The number of final rises of a permutation. St000485The length of the longest cycle of a permutation. St000673The number of non-fixed points of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000081The number of edges of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000654The first descent of a permutation. St000653The last descent of a permutation. St000740The last entry of a permutation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000203The number of external nodes of a binary tree. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001497The position of the largest weak excedence of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000332The positive inversions of an alternating sign matrix. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000843The decomposition number of a perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St000067The inversion number of the alternating sign matrix. St001480The number of simple summands of the module J^2/J^3. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of 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. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. 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. 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. St001645The pebbling number of a connected graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000039The number of crossings of a permutation. St000338The number of pixed points of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000004The major index of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000171The degree of the graph. St000204The number of internal nodes of a binary tree. St000224The sorting index of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000305The inverse major index of a permutation. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000482The (zero)-forcing number of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000794The mak of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001077The prefix exchange distance of a permutation. St001115The number of even descents of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001427The number of descents of a signed permutation. St001428The number of B-inversions of a signed permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000087The number of induced subgraphs. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000542The number of left-to-right-minima of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000990The first ascent of a permutation. St001110The 3-dynamic chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001316The domatic number of a graph. St001330The hat guessing number 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. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace 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. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000094The depth of an ordered tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001180Number of indecomposable injective modules with projective dimension at most 1. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001706The number of closed sets in a graph. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000223The number of nestings in the permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000746The number of pairs with odd minimum in a perfect matching. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000356The number of occurrences of the pattern 13-2. St001083The number of boxed occurrences of 132 in a permutation. St000177The number of free tiles in the pattern. St000178Number of free entries. St001082The number of boxed occurrences of 123 in a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001549The number of restricted non-inversions between exceedances. St001960The number of descents of a permutation minus one if its first entry is not one. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000242The number of indices that are not cyclical small weak excedances. St000317The cycle descent number of a permutation. St000327The number of cover relations in a poset. St000358The number of occurrences of the pattern 31-2. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. 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. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points 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. St001668The number of points of the poset minus the width of the poset. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001948The number of augmented double ascents of a permutation. St000156The Denert index of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001651The Frankl number of a lattice.
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