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Your data matches 237 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 2
[1,1,1]
=> 3
[4]
=> 1
[3,1]
=> 2
[2,2]
=> 2
[2,1,1]
=> 3
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 2
[3,2]
=> 2
[3,1,1]
=> 3
[2,2,1]
=> 3
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 5
[6]
=> 1
[5,1]
=> 2
[4,2]
=> 2
[4,1,1]
=> 3
[3,3]
=> 2
[3,2,1]
=> 3
[3,1,1,1]
=> 4
[2,2,2]
=> 3
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> 6
[7]
=> 1
[6,1]
=> 2
[5,2]
=> 2
[5,1,1]
=> 3
[4,3]
=> 2
[4,2,1]
=> 3
[4,1,1,1]
=> 4
[3,3,1]
=> 3
[3,2,2]
=> 3
[3,2,1,1]
=> 4
[3,1,1,1,1]
=> 5
[2,2,2,1]
=> 4
[2,2,1,1,1]
=> 5
[2,1,1,1,1,1]
=> 6
[1,1,1,1,1,1,1]
=> 7
[8]
=> 1
[7,1]
=> 2
[6,2]
=> 2
[6,1,1]
=> 3
[5,3]
=> 2
[5,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000147
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 100%●distinct values known / distinct values provided: 94%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 100%●distinct values known / distinct values provided: 94%
Values
[1]
=> []
=> []
=> 0 = 1 - 1
[2]
=> []
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[3]
=> []
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[4]
=> []
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[5]
=> []
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
[6]
=> []
=> []
=> 0 = 1 - 1
[5,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[4,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[4,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[3,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[2,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
[7]
=> []
=> []
=> 0 = 1 - 1
[6,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[5,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[5,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[4,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[4,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 6 = 7 - 1
[8]
=> []
=> []
=> 0 = 1 - 1
[7,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[6,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[6,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[5,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[5,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[]
=> ?
=> ?
=> ? = 0 - 1
Description
The largest part of an integer partition.
Matching statistic: St000378
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 98%●distinct values known / distinct values provided: 89%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 98%●distinct values known / distinct values provided: 89%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1 - 2
[2]
=> []
=> ?
=> ?
=> ? = 1 - 2
[1,1]
=> [1]
=> []
=> []
=> 0 = 2 - 2
[3]
=> []
=> ?
=> ?
=> ? = 1 - 2
[2,1]
=> [1]
=> []
=> []
=> 0 = 2 - 2
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[4]
=> []
=> ?
=> ?
=> ? = 1 - 2
[3,1]
=> [1]
=> []
=> []
=> 0 = 2 - 2
[2,2]
=> [2]
=> []
=> []
=> 0 = 2 - 2
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[5]
=> []
=> ?
=> ?
=> ? = 1 - 2
[4,1]
=> [1]
=> []
=> []
=> 0 = 2 - 2
[3,2]
=> [2]
=> []
=> []
=> 0 = 2 - 2
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 5 - 2
[6]
=> []
=> ?
=> ?
=> ? = 1 - 2
[5,1]
=> [1]
=> []
=> []
=> 0 = 2 - 2
[4,2]
=> [2]
=> []
=> []
=> 0 = 2 - 2
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[3,3]
=> [3]
=> []
=> []
=> 0 = 2 - 2
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 3 - 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 5 - 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 4 = 6 - 2
[7]
=> []
=> ?
=> ?
=> ? = 1 - 2
[6,1]
=> [1]
=> []
=> []
=> 0 = 2 - 2
[5,2]
=> [2]
=> []
=> []
=> 0 = 2 - 2
[5,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[4,3]
=> [3]
=> []
=> []
=> 0 = 2 - 2
[4,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[3,3,1]
=> [3,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[3,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 3 - 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 5 - 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [3]
=> 2 = 4 - 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 5 - 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 4 = 6 - 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 5 = 7 - 2
[8]
=> []
=> ?
=> ?
=> ? = 1 - 2
[7,1]
=> [1]
=> []
=> []
=> 0 = 2 - 2
[6,2]
=> [2]
=> []
=> []
=> 0 = 2 - 2
[6,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[5,3]
=> [3]
=> []
=> []
=> 0 = 2 - 2
[5,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[4,4]
=> [4]
=> []
=> []
=> 0 = 2 - 2
[4,3,1]
=> [3,1]
=> [1]
=> [1]
=> 1 = 3 - 2
[4,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 3 - 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 5 - 2
[3,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 3 - 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 4 - 2
[9]
=> []
=> ?
=> ?
=> ? = 1 - 2
[10]
=> []
=> ?
=> ?
=> ? = 1 - 2
[11]
=> []
=> ?
=> ?
=> ? = 1 - 2
[12]
=> []
=> ?
=> ?
=> ? = 1 - 2
[13]
=> []
=> ?
=> ?
=> ? = 1 - 2
[14]
=> []
=> ?
=> ?
=> ? = 1 - 2
[15]
=> []
=> ?
=> ?
=> ? = 1 - 2
[16]
=> []
=> ?
=> ?
=> ? = 1 - 2
[17]
=> []
=> ?
=> ?
=> ? = 1 - 2
[2,2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? = 9 - 2
[2,2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? = 10 - 2
[]
=> ?
=> ?
=> ?
=> ? = 0 - 2
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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 93%●distinct values known / distinct values provided: 56%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 93%●distinct values known / distinct values provided: 56%
Values
[1]
=> 1 => [1] => [1]
=> 1
[2]
=> 0 => [1] => [1]
=> 1
[1,1]
=> 11 => [2] => [2]
=> 2
[3]
=> 1 => [1] => [1]
=> 1
[2,1]
=> 01 => [1,1] => [1,1]
=> 2
[1,1,1]
=> 111 => [3] => [3]
=> 3
[4]
=> 0 => [1] => [1]
=> 1
[3,1]
=> 11 => [2] => [2]
=> 2
[2,2]
=> 00 => [2] => [2]
=> 2
[2,1,1]
=> 011 => [1,2] => [2,1]
=> 3
[1,1,1,1]
=> 1111 => [4] => [4]
=> 4
[5]
=> 1 => [1] => [1]
=> 1
[4,1]
=> 01 => [1,1] => [1,1]
=> 2
[3,2]
=> 10 => [1,1] => [1,1]
=> 2
[3,1,1]
=> 111 => [3] => [3]
=> 3
[2,2,1]
=> 001 => [2,1] => [2,1]
=> 3
[2,1,1,1]
=> 0111 => [1,3] => [3,1]
=> 4
[1,1,1,1,1]
=> 11111 => [5] => [5]
=> 5
[6]
=> 0 => [1] => [1]
=> 1
[5,1]
=> 11 => [2] => [2]
=> 2
[4,2]
=> 00 => [2] => [2]
=> 2
[4,1,1]
=> 011 => [1,2] => [2,1]
=> 3
[3,3]
=> 11 => [2] => [2]
=> 2
[3,2,1]
=> 101 => [1,1,1] => [1,1,1]
=> 3
[3,1,1,1]
=> 1111 => [4] => [4]
=> 4
[2,2,2]
=> 000 => [3] => [3]
=> 3
[2,2,1,1]
=> 0011 => [2,2] => [2,2]
=> 4
[2,1,1,1,1]
=> 01111 => [1,4] => [4,1]
=> 5
[1,1,1,1,1,1]
=> 111111 => [6] => [6]
=> 6
[7]
=> 1 => [1] => [1]
=> 1
[6,1]
=> 01 => [1,1] => [1,1]
=> 2
[5,2]
=> 10 => [1,1] => [1,1]
=> 2
[5,1,1]
=> 111 => [3] => [3]
=> 3
[4,3]
=> 01 => [1,1] => [1,1]
=> 2
[4,2,1]
=> 001 => [2,1] => [2,1]
=> 3
[4,1,1,1]
=> 0111 => [1,3] => [3,1]
=> 4
[3,3,1]
=> 111 => [3] => [3]
=> 3
[3,2,2]
=> 100 => [1,2] => [2,1]
=> 3
[3,2,1,1]
=> 1011 => [1,1,2] => [2,1,1]
=> 4
[3,1,1,1,1]
=> 11111 => [5] => [5]
=> 5
[2,2,2,1]
=> 0001 => [3,1] => [3,1]
=> 4
[2,2,1,1,1]
=> 00111 => [2,3] => [3,2]
=> 5
[2,1,1,1,1,1]
=> 011111 => [1,5] => [5,1]
=> 6
[1,1,1,1,1,1,1]
=> 1111111 => [7] => [7]
=> 7
[8]
=> 0 => [1] => [1]
=> 1
[7,1]
=> 11 => [2] => [2]
=> 2
[6,2]
=> 00 => [2] => [2]
=> 2
[6,1,1]
=> 011 => [1,2] => [2,1]
=> 3
[5,3]
=> 11 => [2] => [2]
=> 2
[5,2,1]
=> 101 => [1,1,1] => [1,1,1]
=> 3
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [11] => [11]
=> ? = 11
[2,1,1,1,1,1,1,1,1,1,1]
=> 01111111111 => [1,10] => [10,1]
=> ? = 11
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [12] => [12]
=> ? = 12
[3,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [11] => [11]
=> ? = 11
[2,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => [2,9] => [9,2]
=> ? = 11
[2,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => ? => ?
=> ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ?
=> ? = 13
[4,1,1,1,1,1,1,1,1,1,1]
=> 01111111111 => [1,10] => [10,1]
=> ? = 11
[3,2,2,1,1,1,1,1,1,1]
=> 1001111111 => ? => ?
=> ? = 10
[3,2,1,1,1,1,1,1,1,1,1]
=> 10111111111 => [1,1,9] => [9,1,1]
=> ? = 11
[3,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [12] => [12]
=> ? = 12
[2,2,2,1,1,1,1,1,1,1,1]
=> 00011111111 => [3,8] => ?
=> ? = 11
[2,2,1,1,1,1,1,1,1,1,1,1]
=> 001111111111 => [2,10] => ?
=> ? = 12
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0111111111111 => ? => ?
=> ? = 13
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => ? => ?
=> ? = 14
[5,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [11] => [11]
=> ? = 11
[4,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => [2,9] => [9,2]
=> ? = 11
[4,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => ? => ?
=> ? = 12
[3,3,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [11] => [11]
=> ? = 11
[3,2,2,2,1,1,1,1,1,1]
=> 1000111111 => ? => ?
=> ? = 10
[3,2,2,1,1,1,1,1,1,1,1]
=> 10011111111 => ? => ?
=> ? = 11
[3,2,1,1,1,1,1,1,1,1,1,1]
=> 101111111111 => ? => ?
=> ? = 12
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ?
=> ? = 13
[2,2,2,2,1,1,1,1,1,1,1]
=> 00001111111 => [4,7] => ?
=> ? = 11
[2,2,2,1,1,1,1,1,1,1,1,1]
=> 000111111111 => ? => ?
=> ? = 12
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> 0011111111111 => ? => ?
=> ? = 13
[2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 01111111111111 => ? => ?
=> ? = 14
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ?
=> ? = 15
[6,1,1,1,1,1,1,1,1,1,1]
=> 01111111111 => [1,10] => [10,1]
=> ? = 11
[5,2,2,1,1,1,1,1,1,1]
=> 1001111111 => ? => ?
=> ? = 10
[5,2,1,1,1,1,1,1,1,1,1]
=> 10111111111 => [1,1,9] => [9,1,1]
=> ? = 11
[5,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [12] => [12]
=> ? = 12
[4,3,1,1,1,1,1,1,1,1,1]
=> 01111111111 => [1,10] => [10,1]
=> ? = 11
[4,2,2,1,1,1,1,1,1,1,1]
=> 00011111111 => [3,8] => ?
=> ? = 11
[4,2,1,1,1,1,1,1,1,1,1,1]
=> 001111111111 => [2,10] => ?
=> ? = 12
[4,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0111111111111 => ? => ?
=> ? = 13
[3,3,2,1,1,1,1,1,1,1,1]
=> 11011111111 => [2,1,8] => ?
=> ? = 11
[3,3,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [12] => [12]
=> ? = 12
[3,2,2,2,1,1,1,1,1,1,1]
=> 10001111111 => ? => ?
=> ? = 11
[3,2,2,1,1,1,1,1,1,1,1,1]
=> 100111111111 => ? => ?
=> ? = 12
[3,2,1,1,1,1,1,1,1,1,1,1,1]
=> 1011111111111 => ? => ?
=> ? = 13
[3,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => ? => ?
=> ? = 14
[2,2,2,2,2,1,1,1,1,1,1]
=> 00000111111 => [5,6] => ?
=> ? = 11
[2,2,2,2,1,1,1,1,1,1,1,1]
=> 000011111111 => [4,8] => ?
=> ? = 12
[2,2,2,1,1,1,1,1,1,1,1,1,1]
=> 0001111111111 => ? => ?
=> ? = 13
[2,2,1,1,1,1,1,1,1,1,1,1,1,1]
=> 00111111111111 => ? => ?
=> ? = 14
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111111 => ? => ?
=> ? = 15
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111111 => ? => ?
=> ? = 16
[7,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [11] => [11]
=> ? = 11
[6,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => [2,9] => [9,2]
=> ? = 11
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001437
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 61% ●values known / values provided: 91%●distinct values known / distinct values provided: 61%
Mp00224: Binary words —runsort⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 61% ●values known / values provided: 91%●distinct values known / distinct values provided: 61%
Values
[1]
=> 1 => 1 => 0 => 1
[2]
=> 0 => 0 => 1 => 1
[1,1]
=> 11 => 11 => 00 => 2
[3]
=> 1 => 1 => 0 => 1
[2,1]
=> 01 => 01 => 10 => 2
[1,1,1]
=> 111 => 111 => 000 => 3
[4]
=> 0 => 0 => 1 => 1
[3,1]
=> 11 => 11 => 00 => 2
[2,2]
=> 00 => 00 => 11 => 2
[2,1,1]
=> 011 => 011 => 100 => 3
[1,1,1,1]
=> 1111 => 1111 => 0000 => 4
[5]
=> 1 => 1 => 0 => 1
[4,1]
=> 01 => 01 => 10 => 2
[3,2]
=> 10 => 01 => 10 => 2
[3,1,1]
=> 111 => 111 => 000 => 3
[2,2,1]
=> 001 => 001 => 110 => 3
[2,1,1,1]
=> 0111 => 0111 => 1000 => 4
[1,1,1,1,1]
=> 11111 => 11111 => 00000 => 5
[6]
=> 0 => 0 => 1 => 1
[5,1]
=> 11 => 11 => 00 => 2
[4,2]
=> 00 => 00 => 11 => 2
[4,1,1]
=> 011 => 011 => 100 => 3
[3,3]
=> 11 => 11 => 00 => 2
[3,2,1]
=> 101 => 011 => 100 => 3
[3,1,1,1]
=> 1111 => 1111 => 0000 => 4
[2,2,2]
=> 000 => 000 => 111 => 3
[2,2,1,1]
=> 0011 => 0011 => 1100 => 4
[2,1,1,1,1]
=> 01111 => 01111 => 10000 => 5
[1,1,1,1,1,1]
=> 111111 => 111111 => 000000 => 6
[7]
=> 1 => 1 => 0 => 1
[6,1]
=> 01 => 01 => 10 => 2
[5,2]
=> 10 => 01 => 10 => 2
[5,1,1]
=> 111 => 111 => 000 => 3
[4,3]
=> 01 => 01 => 10 => 2
[4,2,1]
=> 001 => 001 => 110 => 3
[4,1,1,1]
=> 0111 => 0111 => 1000 => 4
[3,3,1]
=> 111 => 111 => 000 => 3
[3,2,2]
=> 100 => 001 => 110 => 3
[3,2,1,1]
=> 1011 => 0111 => 1000 => 4
[3,1,1,1,1]
=> 11111 => 11111 => 00000 => 5
[2,2,2,1]
=> 0001 => 0001 => 1110 => 4
[2,2,1,1,1]
=> 00111 => 00111 => 11000 => 5
[2,1,1,1,1,1]
=> 011111 => 011111 => 100000 => 6
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 0000000 => 7
[8]
=> 0 => 0 => 1 => 1
[7,1]
=> 11 => 11 => 00 => 2
[6,2]
=> 00 => 00 => 11 => 2
[6,1,1]
=> 011 => 011 => 100 => 3
[5,3]
=> 11 => 11 => 00 => 2
[5,2,1]
=> 101 => 011 => 100 => 3
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 0000000000 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => ? => ? => ? = 11
[3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 0000000000 => ? = 10
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? => ? => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 000000000000 => ? = 12
[3,2,1,1,1,1,1,1,1,1]
=> 1011111111 => ? => ? => ? = 10
[3,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => ? => ? => ? = 11
[2,2,2,1,1,1,1,1,1,1]
=> 0001111111 => ? => ? => ? = 10
[2,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => ? => ? => ? = 11
[2,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => ? => ? => ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ? => ? = 13
[5,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 0000000000 => ? = 10
[4,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? => ? => ? = 10
[3,3,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 0000000000 => ? = 10
[3,2,2,1,1,1,1,1,1,1]
=> 1001111111 => 0011111111 => 1100000000 => ? = 10
[3,2,1,1,1,1,1,1,1,1,1]
=> 10111111111 => ? => ? => ? = 11
[3,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 000000000000 => ? = 12
[2,2,2,2,1,1,1,1,1,1]
=> 0000111111 => ? => ? => ? = 10
[2,2,2,1,1,1,1,1,1,1,1]
=> 00011111111 => ? => ? => ? = 11
[2,2,1,1,1,1,1,1,1,1,1,1]
=> 001111111111 => ? => ? => ? = 12
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0111111111111 => ? => ? => ? = 13
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => ? => ? => ? = 14
[5,2,1,1,1,1,1,1,1,1]
=> 1011111111 => ? => ? => ? = 10
[5,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => ? => ? => ? = 11
[4,2,2,1,1,1,1,1,1,1]
=> 0001111111 => ? => ? => ? = 10
[4,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => ? => ? => ? = 11
[4,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => ? => ? => ? = 12
[3,3,2,1,1,1,1,1,1,1]
=> 1101111111 => ? => ? => ? = 10
[3,3,1,1,1,1,1,1,1,1,1]
=> 11111111111 => ? => ? => ? = 11
[3,2,2,2,1,1,1,1,1,1]
=> 1000111111 => ? => ? => ? = 10
[3,2,2,1,1,1,1,1,1,1,1]
=> 10011111111 => ? => ? => ? = 11
[3,2,1,1,1,1,1,1,1,1,1,1]
=> 101111111111 => ? => ? => ? = 12
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ? => ? = 13
[2,2,2,2,1,1,1,1,1,1,1]
=> 00001111111 => ? => ? => ? = 11
[2,2,2,1,1,1,1,1,1,1,1,1]
=> 000111111111 => ? => ? => ? = 12
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> 0011111111111 => ? => ? => ? = 13
[2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 01111111111111 => ? => ? => ? = 14
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ? => ? = 15
[7,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 0000000000 => ? = 10
[6,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? => ? => ? = 10
[5,3,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 0000000000 => ? = 10
[5,2,2,1,1,1,1,1,1,1]
=> 1001111111 => 0011111111 => 1100000000 => ? = 10
[5,2,1,1,1,1,1,1,1,1,1]
=> 10111111111 => ? => ? => ? = 11
[5,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 000000000000 => ? = 12
[4,4,1,1,1,1,1,1,1,1]
=> 0011111111 => ? => ? => ? = 10
[4,3,2,1,1,1,1,1,1,1]
=> 0101111111 => 0101111111 => 1010000000 => ? = 10
[4,2,2,2,1,1,1,1,1,1]
=> 0000111111 => ? => ? => ? = 10
[4,2,2,1,1,1,1,1,1,1,1]
=> 00011111111 => ? => ? => ? = 11
[4,2,1,1,1,1,1,1,1,1,1,1]
=> 001111111111 => ? => ? => ? = 12
[4,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0111111111111 => ? => ? => ? = 13
Description
The flex of a binary word.
This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
Matching statistic: St000288
(load all 114 compositions to match this statistic)
(load all 114 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 56% ●values known / values provided: 89%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 56% ●values known / values provided: 89%●distinct values known / distinct values provided: 56%
Values
[1]
=> []
=> ?
=> ? => ? = 1 - 2
[2]
=> []
=> ?
=> ? => ? = 1 - 2
[1,1]
=> [1]
=> []
=> => ? = 2 - 2
[3]
=> []
=> ?
=> ? => ? = 1 - 2
[2,1]
=> [1]
=> []
=> => ? = 2 - 2
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 3 - 2
[4]
=> []
=> ?
=> ? => ? = 1 - 2
[3,1]
=> [1]
=> []
=> => ? = 2 - 2
[2,2]
=> [2]
=> []
=> => ? = 2 - 2
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 3 - 2
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[5]
=> []
=> ?
=> ? => ? = 1 - 2
[4,1]
=> [1]
=> []
=> => ? = 2 - 2
[3,2]
=> [2]
=> []
=> => ? = 2 - 2
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 3 - 2
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 3 - 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 5 - 2
[6]
=> []
=> ?
=> ? => ? = 1 - 2
[5,1]
=> [1]
=> []
=> => ? = 2 - 2
[4,2]
=> [2]
=> []
=> => ? = 2 - 2
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 3 - 2
[3,3]
=> [3]
=> []
=> => ? = 2 - 2
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 3 - 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 3 - 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 5 - 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 6 - 2
[7]
=> []
=> ?
=> ? => ? = 1 - 2
[6,1]
=> [1]
=> []
=> => ? = 2 - 2
[5,2]
=> [2]
=> []
=> => ? = 2 - 2
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 3 - 2
[4,3]
=> [3]
=> []
=> => ? = 2 - 2
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 3 - 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 3 - 2
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 3 - 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 5 - 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 2 = 4 - 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 5 - 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 6 - 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 5 = 7 - 2
[8]
=> []
=> ?
=> ? => ? = 1 - 2
[7,1]
=> [1]
=> []
=> => ? = 2 - 2
[6,2]
=> [2]
=> []
=> => ? = 2 - 2
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 3 - 2
[5,3]
=> [3]
=> []
=> => ? = 2 - 2
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 3 - 2
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[4,4]
=> [4]
=> []
=> => ? = 2 - 2
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 3 - 2
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 3 - 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 5 - 2
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 1 = 3 - 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 2 = 4 - 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 5 - 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 6 - 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 2 = 4 - 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 3 = 5 - 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 6 - 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 5 = 7 - 2
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 6 = 8 - 2
[9]
=> []
=> ?
=> ? => ? = 1 - 2
[8,1]
=> [1]
=> []
=> => ? = 2 - 2
[7,2]
=> [2]
=> []
=> => ? = 2 - 2
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 3 - 2
[6,3]
=> [3]
=> []
=> => ? = 2 - 2
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 3 - 2
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[5,4]
=> [4]
=> []
=> => ? = 2 - 2
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 3 - 2
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 3 - 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 4 - 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 5 - 2
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 1 = 3 - 2
[10]
=> []
=> ?
=> ? => ? = 1 - 2
[9,1]
=> [1]
=> []
=> => ? = 2 - 2
[8,2]
=> [2]
=> []
=> => ? = 2 - 2
[7,3]
=> [3]
=> []
=> => ? = 2 - 2
[6,4]
=> [4]
=> []
=> => ? = 2 - 2
[5,5]
=> [5]
=> []
=> => ? = 2 - 2
[11]
=> []
=> ?
=> ? => ? = 1 - 2
[10,1]
=> [1]
=> []
=> => ? = 2 - 2
[9,2]
=> [2]
=> []
=> => ? = 2 - 2
[8,3]
=> [3]
=> []
=> => ? = 2 - 2
[7,4]
=> [4]
=> []
=> => ? = 2 - 2
[6,5]
=> [5]
=> []
=> => ? = 2 - 2
[12]
=> []
=> ?
=> ? => ? = 1 - 2
[11,1]
=> [1]
=> []
=> => ? = 2 - 2
[10,2]
=> [2]
=> []
=> => ? = 2 - 2
[9,3]
=> [3]
=> []
=> => ? = 2 - 2
[8,4]
=> [4]
=> []
=> => ? = 2 - 2
[7,5]
=> [5]
=> []
=> => ? = 2 - 2
[6,6]
=> [6]
=> []
=> => ? = 2 - 2
[13]
=> []
=> ?
=> ? => ? = 1 - 2
[12,1]
=> [1]
=> []
=> => ? = 2 - 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000733
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 56% ●values known / values provided: 87%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 56% ●values known / values provided: 87%●distinct values known / distinct values provided: 56%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1 - 2
[2]
=> []
=> ?
=> ?
=> ? = 1 - 2
[1,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[3]
=> []
=> ?
=> ?
=> ? = 1 - 2
[2,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[4]
=> []
=> ?
=> ?
=> ? = 1 - 2
[3,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[2,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[5]
=> []
=> ?
=> ?
=> ? = 1 - 2
[4,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[3,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 5 - 2
[6]
=> []
=> ?
=> ?
=> ? = 1 - 2
[5,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[4,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[3,3]
=> [3]
=> []
=> []
=> ? = 2 - 2
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 1 = 3 - 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 5 - 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 6 - 2
[7]
=> []
=> ?
=> ?
=> ? = 1 - 2
[6,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[5,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[4,3]
=> [3]
=> []
=> []
=> ? = 2 - 2
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 1 = 3 - 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 5 - 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 4 - 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 5 - 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 6 - 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 7 - 2
[8]
=> []
=> ?
=> ?
=> ? = 1 - 2
[7,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[6,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[5,3]
=> [3]
=> []
=> []
=> ? = 2 - 2
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[4,4]
=> [4]
=> []
=> []
=> ? = 2 - 2
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 1 = 3 - 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 5 - 2
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 1 = 3 - 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 4 - 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 5 - 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 6 - 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 4 - 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 5 - 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 6 - 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 7 - 2
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6 = 8 - 2
[9]
=> []
=> ?
=> ?
=> ? = 1 - 2
[8,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[7,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[6,3]
=> [3]
=> []
=> []
=> ? = 2 - 2
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[5,4]
=> [4]
=> []
=> []
=> ? = 2 - 2
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 1 = 3 - 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 4 - 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 5 - 2
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 1 = 3 - 2
[10]
=> []
=> ?
=> ?
=> ? = 1 - 2
[9,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[8,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[7,3]
=> [3]
=> []
=> []
=> ? = 2 - 2
[6,4]
=> [4]
=> []
=> []
=> ? = 2 - 2
[5,5]
=> [5]
=> []
=> []
=> ? = 2 - 2
[11]
=> []
=> ?
=> ?
=> ? = 1 - 2
[10,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[9,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[8,3]
=> [3]
=> []
=> []
=> ? = 2 - 2
[7,4]
=> [4]
=> []
=> []
=> ? = 2 - 2
[6,5]
=> [5]
=> []
=> []
=> ? = 2 - 2
[12]
=> []
=> ?
=> ?
=> ? = 1 - 2
[11,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
[10,2]
=> [2]
=> []
=> []
=> ? = 2 - 2
[9,3]
=> [3]
=> []
=> []
=> ? = 2 - 2
[8,4]
=> [4]
=> []
=> []
=> ? = 2 - 2
[7,5]
=> [5]
=> []
=> []
=> ? = 2 - 2
[6,6]
=> [6]
=> []
=> []
=> ? = 2 - 2
[13]
=> []
=> ?
=> ?
=> ? = 1 - 2
[12,1]
=> [1]
=> []
=> []
=> ? = 2 - 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000734
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 61% ●values known / values provided: 78%●distinct values known / distinct values provided: 61%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 61% ●values known / values provided: 78%●distinct values known / distinct values provided: 61%
Values
[1]
=> []
=> []
=> []
=> ? = 1 - 1
[2]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[3]
=> []
=> []
=> []
=> ? = 1 - 1
[2,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[1,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2 = 3 - 1
[4]
=> []
=> []
=> []
=> ? = 1 - 1
[3,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[2,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3 = 4 - 1
[5]
=> []
=> []
=> []
=> ? = 1 - 1
[4,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[3,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2 = 3 - 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4 = 5 - 1
[6]
=> []
=> []
=> []
=> ? = 1 - 1
[5,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[4,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[4,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2 = 3 - 1
[3,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3 = 4 - 1
[2,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 6 - 1
[7]
=> []
=> []
=> []
=> ? = 1 - 1
[6,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[5,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[5,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2 = 3 - 1
[4,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 2 - 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3 = 4 - 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4 = 5 - 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 5 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 6 = 7 - 1
[8]
=> []
=> []
=> []
=> ? = 1 - 1
[7,1]
=> [1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[6,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[6,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2 = 3 - 1
[5,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 2 - 1
[5,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3 = 4 - 1
[4,4]
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 2 - 1
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
[4,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4 = 5 - 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
[9]
=> []
=> []
=> []
=> ? = 1 - 1
[10]
=> []
=> []
=> []
=> ? = 1 - 1
[11]
=> []
=> []
=> []
=> ? = 1 - 1
[12]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 12 - 1
[13]
=> []
=> []
=> []
=> ? = 1 - 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 7 - 1
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 8 - 1
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 9 - 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 10 - 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 11 - 1
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 12 - 1
[14]
=> []
=> []
=> []
=> ? = 1 - 1
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 7 - 1
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? = 8 - 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 6 - 1
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 7 - 1
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? = 8 - 1
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 9 - 1
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? = 10 - 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 7 - 1
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 8 - 1
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 9 - 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 10 - 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 11 - 1
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 12 - 1
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 8 - 1
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? = 9 - 1
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? = 10 - 1
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? = 11 - 1
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [11,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12]]
=> ? = 12 - 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? = 14 - 1
[15]
=> []
=> []
=> []
=> ? = 1 - 1
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? = 7 - 1
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? = 7 - 1
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? = 8 - 1
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? = 9 - 1
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 7 - 1
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? = 8 - 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 6 - 1
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 7 - 1
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? = 8 - 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000676
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> []
=> []
=> 0 = 1 - 1
[2]
=> []
=> []
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[3]
=> []
=> []
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[1,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[4]
=> []
=> []
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[2,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[5]
=> []
=> []
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[3,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[6]
=> []
=> []
=> []
=> 0 = 1 - 1
[5,1]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[4,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[2,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[7]
=> []
=> []
=> []
=> 0 = 1 - 1
[6,1]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[5,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[5,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[8]
=> []
=> []
=> []
=> 0 = 1 - 1
[7,1]
=> [1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[6,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[6,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[5,2,1]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10 - 1
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9 - 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9 - 1
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10 - 1
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9 - 1
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10 - 1
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12 - 1
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7 - 1
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9 - 1
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10 - 1
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 8 - 1
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9 - 1
[3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10 - 1
[3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9 - 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 10 - 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 11 - 1
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12 - 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 13 - 1
[5,5,1,1,1,1]
=> [5,1,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[5,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7 - 1
[5,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[5,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9 - 1
[5,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10 - 1
[4,4,2,1,1,1,1]
=> [4,2,1,1,1,1]
=> [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 7 - 1
[4,4,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 8 - 1
[4,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 8 - 1
[4,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[4,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9 - 1
[4,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10 - 1
[4,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 8 - 1
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 8 - 1
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 9 - 1
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 10 - 1
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 9 - 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 10 - 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 11 - 1
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12 - 1
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 9 - 1
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 10 - 1
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 11 - 1
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 12 - 1
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 13 - 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000507
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Values
[1]
=> []
=> []
=> []
=> 0 = 1 - 1
[2]
=> []
=> []
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[3]
=> []
=> []
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
[4]
=> []
=> []
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[2,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
[5]
=> []
=> []
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[3,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
[6]
=> []
=> []
=> []
=> 0 = 1 - 1
[5,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[4,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 3 - 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5 = 6 - 1
[7]
=> []
=> []
=> []
=> 0 = 1 - 1
[6,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[5,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 3 - 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4 = 5 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 6 = 7 - 1
[8]
=> []
=> []
=> []
=> 0 = 1 - 1
[7,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
[6,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2 = 3 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 12 - 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 7 - 1
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ?
=> ? = 8 - 1
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> [[1,2,3,4,5,6,8,10],[7,9,11]]
=> ? = 9 - 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ?
=> ? = 10 - 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 11 - 1
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 12 - 1
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 5 - 1
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? = 6 - 1
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 6 - 1
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 7 - 1
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> [[1,2,3,4,5,6,9],[7,10],[8,11]]
=> ? = 8 - 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 6 - 1
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 7 - 1
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ?
=> ? = 8 - 1
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ?
=> ? = 9 - 1
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? = 10 - 1
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 7 - 1
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ?
=> ? = 8 - 1
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> [[1,2,3,4,5,6,8,10],[7,9,11]]
=> ? = 9 - 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ?
=> ? = 10 - 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 11 - 1
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 12 - 1
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ?
=> ? = 8 - 1
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? = 9 - 1
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ?
=> ? = 10 - 1
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? = 11 - 1
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 12 - 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? = 14 - 1
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 4 - 1
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> ? = 5 - 1
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [[1,2,3,4,8],[5,9],[6,10],[7,11]]
=> ? = 6 - 1
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? = 5 - 1
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? = 5 - 1
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> ? = 6 - 1
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ?
=> ? = 7 - 1
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9],[10],[11]]
=> ? = 6 - 1
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ?
=> ? = 7 - 1
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ?
=> ? = 8 - 1
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ?
=> ? = 9 - 1
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 5 - 1
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? = 6 - 1
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 6 - 1
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 7 - 1
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> [[1,2,3,4,5,6,9],[7,10],[8,11]]
=> ? = 8 - 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 6 - 1
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 7 - 1
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ?
=> ? = 8 - 1
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ?
=> ? = 9 - 1
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? = 10 - 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
The following 227 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000007The number of saliances of the permutation. St001777The number of weak descents in an integer composition. 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000006The dinv of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000738The first entry in the last row of a standard tableau. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000809The reduced reflection length of the permutation. St001462The number of factors of a standard tableaux under concatenation. St000012The area of a Dyck path. St000015The number of peaks of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000216The absolute length of a permutation. St000839The largest opener of a set partition. St000105The number of blocks in the set partition. St000925The number of topologically connected components of a set partition. St000528The height of a poset. St001152The number of pairs with even minimum in a perfect matching. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000080The rank of the poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000325The width of the tree associated to a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000354The number of recoils 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. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000013The height of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000691The number of changes of a binary word. St000011The number of touch points (or returns) of a Dyck path. St001497The position of the largest weak excedence of a permutation. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000439The position of the first down step of a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000019The cardinality of the support of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000740The last entry of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000306The bounce count of a Dyck path. St001432The order dimension of the partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000956The maximal displacement of a permutation. St000382The first part of an integer composition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001389The number of partitions of the same length below the given integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001050The number of terminal closers of a set partition. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001933The largest multiplicity of a part in an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000067The inversion number of the alternating sign matrix. St000167The number of leaves of an ordered tree. St000204The number of internal nodes of a binary tree. St000548The number of different non-empty partial sums of an integer partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000051The size of the left subtree of a binary tree. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000356The number of occurrences of the pattern 13-2. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000444The length of the maximal rise of a Dyck path. St000874The position of the last double rise in a Dyck path. St000025The number of initial rises of a Dyck path. St000820The number of compositions obtained by rotating the composition. 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. 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:
St000024The number of double up and double down steps of a Dyck path. St000211The rank of the set partition. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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 $eA$. 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. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000031The number of cycles in the cycle decomposition of a permutation. St000381The largest part of an integer composition. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000912The number of maximal antichains in a poset. St000808The number of up steps of the associated bargraph. St000829The Ulam distance of a permutation to the identity permutation. St000446The disorder of a permutation. St000153The number of adjacent cycles of a permutation. St000160The multiplicity of the smallest part of a partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000993The multiplicity of the largest part of an integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000292The number of ascents of a binary word. St000702The number of weak deficiencies of a permutation. St000097The order of the largest clique of the graph. St000383The last part of an integer composition. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000662The staircase size of the code of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000098The chromatic number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000990The first ascent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000168The number of internal nodes of an ordered tree. St001153The number of blocks with even minimum in a set partition. St001180Number of indecomposable injective modules with projective dimension at most 1. 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 $eA$. 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. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001581The achromatic number of a graph. St000083The number of left oriented leafs of a binary tree except the first one. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000840The number of closers smaller than the largest opener in a perfect matching. St000069The number of maximal elements of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000068The number of minimal elements in a poset. St000527The width of the poset. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000632The jump number of the poset. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000172The Grundy number of a graph. St000722The number of different neighbourhoods in a graph. St001029The size of the core 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. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000133The "bounce" of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001963The tree-depth of a graph. St000868The aid statistic in the sense of Shareshian-Wachs. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. 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. St000746The number of pairs with odd minimum in a perfect matching. St001427The number of descents of a signed permutation. St000308The height of the tree associated to a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001962The proper pathwidth of a graph. St000719The number of alignments in a perfect matching. St000989The number of final rises of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000822The Hadwiger number of the graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St000021The number of descents of a permutation. St000061The number of nodes on the left branch of a binary tree. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001812The biclique partition number of a graph. St001644The dimension of a graph. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation.
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