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Your data matches 218 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000147: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 3
[3,1,1]
=> 3
[2,2,1]
=> 2
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 4
[4,1,1]
=> 4
[3,3]
=> 3
[3,2,1]
=> 3
[3,1,1,1]
=> 3
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 7
[6,1]
=> 6
[5,2]
=> 5
[5,1,1]
=> 5
[4,3]
=> 4
[4,2,1]
=> 4
[4,1,1,1]
=> 4
[3,3,1]
=> 3
[3,2,2]
=> 3
[3,2,1,1]
=> 3
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 7
[6,2]
=> 6
[6,1,1]
=> 6
[5,3]
=> 5
[5,2,1]
=> 5
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 2
[1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> 3
[2,1]
=> [2,1]
=> 2
[1,1,1]
=> [3]
=> 1
[4]
=> [1,1,1,1]
=> 4
[3,1]
=> [2,1,1]
=> 3
[2,2]
=> [2,2]
=> 2
[2,1,1]
=> [3,1]
=> 2
[1,1,1,1]
=> [4]
=> 1
[5]
=> [1,1,1,1,1]
=> 5
[4,1]
=> [2,1,1,1]
=> 4
[3,2]
=> [2,2,1]
=> 3
[3,1,1]
=> [3,1,1]
=> 3
[2,2,1]
=> [3,2]
=> 2
[2,1,1,1]
=> [4,1]
=> 2
[1,1,1,1,1]
=> [5]
=> 1
[6]
=> [1,1,1,1,1,1]
=> 6
[5,1]
=> [2,1,1,1,1]
=> 5
[4,2]
=> [2,2,1,1]
=> 4
[4,1,1]
=> [3,1,1,1]
=> 4
[3,3]
=> [2,2,2]
=> 3
[3,2,1]
=> [3,2,1]
=> 3
[3,1,1,1]
=> [4,1,1]
=> 3
[2,2,2]
=> [3,3]
=> 2
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 2
[1,1,1,1,1,1]
=> [6]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [2,1,1,1,1,1]
=> 6
[5,2]
=> [2,2,1,1,1]
=> 5
[5,1,1]
=> [3,1,1,1,1]
=> 5
[4,3]
=> [2,2,2,1]
=> 4
[4,2,1]
=> [3,2,1,1]
=> 4
[4,1,1,1]
=> [4,1,1,1]
=> 4
[3,3,1]
=> [3,2,2]
=> 3
[3,2,2]
=> [3,3,1]
=> 3
[3,2,1,1]
=> [4,2,1]
=> 3
[3,1,1,1,1]
=> [5,1,1]
=> 3
[2,2,2,1]
=> [4,3]
=> 2
[2,2,1,1,1]
=> [5,2]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [2,1,1,1,1,1,1]
=> 7
[6,2]
=> [2,2,1,1,1,1]
=> 6
[6,1,1]
=> [3,1,1,1,1,1]
=> 6
[5,3]
=> [2,2,2,1,1]
=> 5
[5,2,1]
=> [3,2,1,1,1]
=> 5
Description
The length of the partition.
Matching statistic: St000378
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00322: Integer partitions āLoehr-Warringtonā¶ Integer partitions
St000378: Integer partitions ā¶ ā¤Result quality: 89% āvalues known / values provided: 97%ādistinct values known / distinct values provided: 89%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00322: Integer partitions āLoehr-Warringtonā¶ Integer partitions
St000378: Integer partitions ā¶ ā¤Result quality: 89% āvalues known / values provided: 97%ādistinct values known / distinct values provided: 89%
Values
[1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[2]
=> [1,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> []
=> 0 = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,1]
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> []
=> 0 = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> []
=> 0 = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [3]
=> 2 = 3 - 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> 0 = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 5 = 6 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 3 = 4 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [4]
=> 2 = 3 - 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [3]
=> 2 = 3 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> 0 = 1 - 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 6 = 7 - 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 5 = 6 - 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 4 = 5 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 3 = 4 - 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [2,2]
=> 3 = 4 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [4]
=> 2 = 3 - 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [3]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> 0 = 1 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> 7 = 8 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 6 = 7 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,2]
=> 5 = 6 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [4,1,1]
=> 4 = 5 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 4 = 5 - 1
[9,6]
=> [2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? = 9 - 1
[8,7]
=> [2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? = 8 - 1
[9,6,1]
=> [3,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? = 9 - 1
[8,7,1]
=> [3,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? = 8 - 1
[6,5,5]
=> [3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> ? = 6 - 1
[17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [6,4,3,2,1]
=> ? = 17 - 1
[15,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [6,4,3,2]
=> ? = 15 - 1
[13,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1,1,1]
=> [6,4,2,2,1]
=> ? = 13 - 1
[13,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [6,4,3,1]
=> ? = 13 - 1
[12,5]
=> [2,2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1,1,1]
=> [6,3,3,2,1]
=> ? = 12 - 1
[11,6]
=> [2,2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1,1]
=> [5,3,3,3,1]
=> ? = 11 - 1
[11,4,2]
=> [3,3,2,2,1,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [6,3,3,2]
=> ? = 11 - 1
[11,3,3]
=> [3,3,3,1,1,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1,1,1]
=> [6,4,2,1,1]
=> ? = 11 - 1
[10,4,3]
=> [3,3,3,2,1,1,1,1,1,1]
=> [3,3,2,1,1,1,1,1,1]
=> [6,3,3,1,1]
=> ? = 10 - 1
[10,3,2,2]
=> [4,4,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> [6,3,3,1]
=> ? = 10 - 1
[9,6,1,1]
=> [4,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? = 9 - 1
[9,4,4]
=> [3,3,3,3,1,1,1,1,1]
=> [3,3,3,1,1,1,1,1]
=> [6,3,2,2,1]
=> ? = 9 - 1
[9,4,2,2]
=> [4,4,2,2,1,1,1,1,1]
=> [4,2,2,1,1,1,1,1]
=> [7,3,2,1]
=> ? = 9 - 1
[8,7,2]
=> [3,3,2,2,2,2,2,1]
=> [3,2,2,2,2,2,1]
=> [4,3,3,3,1]
=> ? = 8 - 1
[8,7,1,1]
=> [4,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? = 8 - 1
[8,5,4]
=> [3,3,3,3,2,1,1,1]
=> [3,3,3,2,1,1,1]
=> [4,4,2,2,2]
=> ? = 8 - 1
[8,5,2,2]
=> [4,4,2,2,2,1,1,1]
=> [4,2,2,2,1,1,1]
=> [5,5,1,1,1]
=> ? = 8 - 1
[8,4,3,2]
=> [4,4,3,2,1,1,1,1]
=> [4,3,2,1,1,1,1]
=> [6,3,2,1,1]
=> ? = 8 - 1
[7,6,4]
=> [3,3,3,3,2,2,1]
=> [3,3,3,2,2,1]
=> [7,6,1]
=> ? = 7 - 1
[7,6,2,2]
=> [4,4,2,2,2,2,1]
=> [4,2,2,2,2,1]
=> [5,2,2,2,2]
=> ? = 7 - 1
[7,5,5]
=> [3,3,3,3,3,1,1]
=> [3,3,3,3,1,1]
=> [7,3,1,1,1,1]
=> ? = 7 - 1
[7,4,4,2]
=> [4,4,3,3,1,1,1]
=> [4,3,3,1,1,1]
=> [4,3,2,2,2]
=> ? = 7 - 1
[6,5,5,1]
=> [4,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> ? = 6 - 1
[6,5,4,2]
=> [4,4,3,3,2,1]
=> [4,3,3,2,1]
=> [7,2,2,2]
=> ? = 6 - 1
[6,4,4,3]
=> [4,4,4,3,1,1]
=> [4,4,3,1,1]
=> [4,2,2,2,2,1]
=> ? = 6 - 1
[5,5,5,2]
=> [4,4,3,3,3]
=> [4,3,3,3]
=> [7,1,1,1,1,1,1]
=> ? = 5 - 1
[]
=> []
=> ?
=> ?
=> ? = 0 - 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000288
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St000288: Binary words ā¶ ā¤Result quality: 56% āvalues known / values provided: 80%ādistinct values known / distinct values provided: 56%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St000288: Binary words ā¶ ā¤Result quality: 56% āvalues known / values provided: 80%ādistinct values known / distinct values provided: 56%
Values
[1]
=> [1]
=> []
=> => ? = 1 - 1
[2]
=> [1,1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1]
=> [2]
=> []
=> => ? = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
[2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> => ? = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
[2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> => ? = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> 1010 => 2 = 3 - 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
[2,2,1]
=> [3,2]
=> [2]
=> 100 => 1 = 2 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> => ? = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 5 = 6 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 3 = 4 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
[3,3]
=> [2,2,2]
=> [2,2]
=> 1100 => 2 = 3 - 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 1010 => 2 = 3 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
[2,2,2]
=> [3,3]
=> [3]
=> 1000 => 1 = 2 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> 100 => 1 = 2 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> []
=> => ? = 1 - 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 6 = 7 - 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 5 = 6 - 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 4 = 5 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 11010 => 3 = 4 - 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => 3 = 4 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 1100 => 2 = 3 - 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 10010 => 2 = 3 - 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 1010 => 2 = 3 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
[2,2,2,1]
=> [4,3]
=> [3]
=> 1000 => 1 = 2 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 100 => 1 = 2 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 10 => 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> => ? = 1 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7 = 8 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 6 = 7 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 5 = 6 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 4 = 5 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 4 = 5 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 11100 => 3 = 4 - 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> 11010 => 3 = 4 - 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 100110 => 3 = 4 - 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 10110 => 3 = 4 - 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> 10100 => 2 = 3 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> => ? = 1 - 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> => ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> => ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> 111111111110 => ? = 12 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> => ? = 1 - 1
[13]
=> [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]
=> 1111111111110 => ? = 13 - 1
[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]
=> 111111111110 => ? = 12 - 1
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 11 - 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> []
=> => ? = 1 - 1
[14]
=> [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]
=> 11111111111110 => ? = 14 - 1
[13,1]
=> [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]
=> 1111111111110 => ? = 13 - 1
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 12 - 1
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 12 - 1
[11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 11 - 1
[11,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 11 - 1
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 10 - 1
[10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? = 10 - 1
[9,5]
=> [2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 9 - 1
[8,6]
=> [2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 8 - 1
[8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? = 8 - 1
[8,3,3]
=> [3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 8 - 1
[7,5,2]
=> [3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> 101110110 => ? = 7 - 1
[7,4,3]
=> [3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> 110101110 => ? = 7 - 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [14]
=> []
=> => ? = 1 - 1
[15]
=> [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]
=> 111111111111110 => ? = 15 - 1
[14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111110 => ? = 14 - 1
[13,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> 10111111111110 => ? = 13 - 1
[13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 13 - 1
[12,3]
=> [2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> 1101111111110 => ? = 12 - 1
[12,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? = 12 - 1
[12,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 12 - 1
[11,4]
=> [2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> 111011111110 => ? = 11 - 1
[11,3,1]
=> [3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 11 - 1
[11,2,2]
=> [3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> 1001111111110 => ? = 11 - 1
[11,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 11 - 1
[10,5]
=> [2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> 11110111110 => ? = 10 - 1
[10,4,1]
=> [3,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 10 - 1
[10,3,2]
=> [3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> 101011111110 => ? = 10 - 1
[10,2,2,1]
=> [4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? = 10 - 1
[9,6]
=> [2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> 1111101110 => ? = 9 - 1
[9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 9 - 1
[9,4,2]
=> [3,3,2,2,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> 10110111110 => ? = 9 - 1
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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
St000733: Standard tableaux ā¶ ā¤Result quality: 56% āvalues known / values provided: 75%ādistinct values known / distinct values provided: 56%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
St000733: Standard tableaux ā¶ ā¤Result quality: 56% āvalues known / values provided: 75%ādistinct values known / distinct values provided: 56%
Values
[1]
=> [1]
=> []
=> []
=> ? = 1 - 1
[2]
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> []
=> ? = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
[2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
[2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
[2,2,1]
=> [3,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 4 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
[2,2,2]
=> [3,3]
=> [3]
=> [[1,2,3]]
=> 1 = 2 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1 - 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6 = 7 - 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4 = 5 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3 = 4 - 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 4 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [[1,2,3]]
=> 1 = 2 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7 = 8 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6 = 7 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 5 = 6 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 4 = 5 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4 = 5 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 3 = 4 - 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3 = 4 - 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 4 - 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2 = 3 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 12 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 1 - 1
[13]
=> [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
[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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 12 - 1
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11 - 1
[10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 - 1
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 9 - 1
[8,5]
=> [2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 8 - 1
[7,6]
=> [2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 7 - 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> []
=> []
=> ? = 1 - 1
[14]
=> [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]]
=> ? = 14 - 1
[13,1]
=> [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]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 13 - 1
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
[12,1,1]
=> [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]]
=> ? = 12 - 1
[11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11 - 1
[11,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11 - 1
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 10 - 1
[10,3,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 - 1
[10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10 - 1
[9,5]
=> [2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 9 - 1
[9,4,1]
=> [3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 9 - 1
[9,3,2]
=> [3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 9 - 1
[8,6]
=> [2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 8 - 1
[8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 8 - 1
[8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? = 8 - 1
[8,3,3]
=> [3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? = 8 - 1
[7,6,1]
=> [3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 7 - 1
[7,5,2]
=> [3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 7 - 1
[7,4,3]
=> [3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 7 - 1
[6,6,2]
=> [3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 6 - 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [14]
=> []
=> []
=> ? = 1 - 1
[15]
=> [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]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
=> ? = 15 - 1
[14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 14 - 1
[13,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 13 - 1
[13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 13 - 1
[12,3]
=> [2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 12 - 1
[12,2,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
[12,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 12 - 1
[11,4]
=> [2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 11 - 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000645
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00132: Dyck paths āswitch returns and last double riseā¶ Dyck paths
St000645: Dyck paths ā¶ ā¤Result quality: 56% āvalues known / values provided: 56%ādistinct values known / distinct values provided: 67%
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00132: Dyck paths āswitch returns and last double riseā¶ Dyck paths
St000645: Dyck paths ā¶ ā¤Result quality: 56% āvalues known / values provided: 56%ādistinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> 8
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> 7
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 6
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> ? = 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 9
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 8
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0]
=> ? = 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> ? = 3
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 3
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? = 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 12
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 11
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 9
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 9
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? = 8
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 8
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0]
=> ? = 5
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0]
=> ? = 4
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0]
=> ? = 4
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> ? = 3
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> ? = 3
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 3
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ?
=> ? = 3
[2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 2
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? = 1
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 13
[12,1]
=> [1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 12
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 11
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 11
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 10
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 10
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 9
[9,3,1]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 9
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 9
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 9
[8,5]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> ? = 8
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? = 8
[8,3,2]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 8
[8,3,1,1]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 8
[6,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0]
=> ? = 6
[5,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0]
=> ? = 5
[5,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,0]
=> ? = 5
[4,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0]
=> ? = 4
[4,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0]
=> ? = 4
[4,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,0]
=> ? = 4
[4,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ?
=> ? = 4
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St001227
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001227: Dyck paths ā¶ ā¤Result quality: 28% āvalues known / values provided: 55%ādistinct values known / distinct values provided: 28%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001227: Dyck paths ā¶ ā¤Result quality: 28% āvalues known / values provided: 55%ādistinct values known / distinct values provided: 28%
Values
[1]
=> [1]
=> []
=> []
=> ? = 1 - 1
[2]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> []
=> ? = 1 - 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1 - 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1 - 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1 - 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1 - 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 5 = 6 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[3,3,1,1]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2,2,1]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,1,1]
=> [5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1 - 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 - 1
[7,2]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 7 - 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 1 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 10 - 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,2]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 8 - 1
[8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 - 1
[7,3]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 7 - 1
[7,2,1]
=> [3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 7 - 1
[7,1,1,1]
=> [4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 1 - 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 10 - 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 9 - 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,3]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 8 - 1
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 8 - 1
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 - 1
[7,4]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[7,3,1]
=> [3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 7 - 1
[7,2,2]
=> [3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 7 - 1
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 7 - 1
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 10 - 1
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 10 - 1
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 9 - 1
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 9 - 1
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,4]
=> [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 8 - 1
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 8 - 1
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 8 - 1
[8,2,1,1]
=> [4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 8 - 1
[8,1,1,1,1]
=> [5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 8 - 1
[7,5]
=> [2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000738
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00059: Permutations āRobinson-Schensted insertion tableauā¶ Standard tableaux
St000738: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 53%ādistinct values known / distinct values provided: 50%
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00059: Permutations āRobinson-Schensted insertion tableauā¶ Standard tableaux
St000738: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 53%ādistinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[1,2],[3]]
=> 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[1,3],[2],[4]]
=> 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[1,2,3,4,5],[6]]
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[1,3,4],[2],[5]]
=> 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 4 = 3 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 4 = 3 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7]]
=> 7 = 6 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [[1,3,4,5],[2],[6]]
=> 6 = 5 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[1,2,4],[3],[5]]
=> 5 = 4 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[1,3,4],[2],[5]]
=> 5 = 4 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 4 = 3 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4 = 3 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,5],[2],[4]]
=> 4 = 3 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 3 = 2 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [[1,4,5,6],[2],[3]]
=> 3 = 2 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[1,3,4,5,6,7],[2]]
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7],[8]]
=> 8 = 7 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [[1,3,4,5,6],[2],[7]]
=> 7 = 6 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [[1,2,4,5],[3],[6]]
=> 6 = 5 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [[1,3,4,5],[2],[6]]
=> 6 = 5 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 5 = 4 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[1,4],[2],[3],[5]]
=> 5 = 4 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5 = 4 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[1,2],[3,5],[4]]
=> 4 = 3 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [[1,3,5,6],[2],[4]]
=> 4 = 3 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [[1,4,5,6],[2],[3]]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [[1,4,5,6,7],[2],[3]]
=> 3 = 2 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[1,3,4,5,6,7,8],[2]]
=> 2 = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [[1,2,3,4,5,6,7,8],[9]]
=> 9 = 8 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [[1,3,4,5,6,7],[2],[8]]
=> 8 = 7 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [[1,2,4,5,6],[3],[7]]
=> 7 = 6 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [[1,3,4,5,6],[2],[7]]
=> 7 = 6 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[1,2,3,5],[4],[6]]
=> 6 = 5 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [[1,4,5],[2],[3],[6]]
=> 6 = 5 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 9 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1 + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11],[12]]
=> ? = 11 + 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10],[2],[11]]
=> ? = 10 + 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => [[1,2,4,5,6,7,8,9],[3],[10]]
=> ? = 9 + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 9 + 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => ?
=> ? = 8 + 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,1,5,6,7] => ?
=> ? = 7 + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [5,2,3,4,6,7,8,9,1] => ?
=> ? = 4 + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => [[1,3,5,6,7,8,9,10],[2],[4]]
=> ? = 3 + 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ?
=> ? = 2 + 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ?
=> ? = 2 + 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 12 + 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2],[12]]
=> ? = 11 + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [11,3,1,2,4,5,6,7,8,9,10] => [[1,2,4,5,6,7,8,9,10],[3],[11]]
=> ? = 10 + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,3,1,4,5,6,7,8,9,10] => ?
=> ? = 10 + 1
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [10,4,1,2,3,5,6,7,8,9] => [[1,2,3,5,6,7,8,9],[4],[10]]
=> ? = 9 + 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,2,1,4,5,6,7,8,9] => ?
=> ? = 9 + 1
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,1,5,6,7,8,9] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 9 + 1
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [9,5,1,2,3,4,6,7,8] => ?
=> ? = 8 + 1
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [9,4,2,1,3,5,6,7,8] => ?
=> ? = 8 + 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,1,2,5,6,7,8] => [[1,2,5,6,7,8],[3,4],[9]]
=> ? = 8 + 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,4,1,5,6,7,8] => ?
=> ? = 8 + 1
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 7 + 1
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 7 + 1
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,5,1,6,7] => ?
=> ? = 7 + 1
[5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,3,2,4,5,7,8,1] => ?
=> ? = 5 + 1
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [6,2,3,4,5,7,8,9,1] => ?
=> ? = 5 + 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [5,3,2,4,6,7,8,9,1] => ?
=> ? = 4 + 1
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [5,2,3,4,6,7,8,9,10,1] => ?
=> ? = 4 + 1
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [4,3,5,2,6,7,8,9,1] => ?
=> ? = 3 + 1
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,11,1] => ?
=> ? = 3 + 1
[2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,9,1] => ?
=> ? = 2 + 1
[2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,10,1] => ?
=> ? = 2 + 1
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,11,1] => ?
=> ? = 2 + 1
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,12,1] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 13 + 1
[12,1]
=> [1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 12 + 1
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 11 + 1
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 11 + 1
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 10 + 1
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [11,3,2,1,4,5,6,7,8,9,10] => ?
=> ? = 10 + 1
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 10 + 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [10,5,1,2,3,4,6,7,8,9] => ?
=> ? = 9 + 1
[9,3,1]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [10,4,2,1,3,5,6,7,8,9] => ?
=> ? = 9 + 1
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 9 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000734
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
Mp00070: Permutations āRobinson-Schensted recording tableauā¶ Standard tableaux
St000734: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 52%ādistinct values known / distinct values provided: 50%
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
Mp00070: Permutations āRobinson-Schensted recording tableauā¶ Standard tableaux
St000734: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 52%ādistinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [[1,6],[2],[3],[4],[5]]
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 4 = 3 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 4 = 3 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [[1,7],[2],[3],[4],[5],[6]]
=> 7 = 6 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [[1,2,6],[3],[4],[5]]
=> 6 = 5 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 5 = 4 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 5 = 4 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 4 = 3 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 4 = 3 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3 = 2 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7]]
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [[1,8],[2],[3],[4],[5],[6],[7]]
=> 8 = 7 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [[1,2,7],[3],[4],[5],[6]]
=> 7 = 6 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [[1,3,6],[2],[4],[5]]
=> 6 = 5 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [[1,2,6],[3],[4],[5]]
=> 6 = 5 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 5 = 4 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 5 = 4 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 5 = 4 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 4 = 3 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 4 = 3 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4 = 3 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [[1,2,4],[3],[5],[6]]
=> 4 = 3 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [[1,2,3],[4],[5],[6],[7]]
=> 3 = 2 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8]]
=> 2 = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> 9 = 8 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [[1,2,8],[3],[4],[5],[6],[7]]
=> 8 = 7 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [[1,3,7],[2],[4],[5],[6]]
=> 7 = 6 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [[1,2,7],[3],[4],[5],[6]]
=> 7 = 6 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [[1,4,6],[2],[3],[5]]
=> 6 = 5 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [[1,2,3,6],[4],[5]]
=> 6 = 5 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 10 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
=> ? = 8 + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9]]
=> ? = 3 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,10,9,8,7,6,5,4,3,2,1,12] => [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11 + 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,10,8,7,6,5,4,3,2,1,11] => ?
=> ? = 10 + 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,7,9,6,5,4,3,2,1,10] => [[1,3,10],[2],[4],[5],[6],[7],[8],[9]]
=> ? = 9 + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,9,8,6,5,4,3,2,1,10] => ?
=> ? = 9 + 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [7,6,5,8,4,3,2,1,9] => [[1,4,9],[2],[3],[5],[6],[7],[8]]
=> ? = 8 + 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,8,7,6,4,3,2,1,9] => ?
=> ? = 8 + 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,7,6,5,4,2,1,8] => ?
=> ? = 7 + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,8,7,6,9,5,4,3,2] => ?
=> ? = 4 + 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,5,7,8,6,4,3,2] => ?
=> ? = 3 + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,9,8,10,7,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9],[10]]
=> ? = 3 + 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,6,9,8,7,5,4,3,2] => ?
=> ? = 2 + 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,8,10,9,7,6,5,4,3,2] => ?
=> ? = 2 + 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10,11,9,8,7,6,5,4,3,2] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 12 + 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,11,9,8,7,6,5,4,3,2,1,12] => ?
=> ? = 11 + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,10,7,6,5,4,3,2,1,11] => [[1,3,11],[2],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 10 + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [8,10,9,7,6,5,4,3,2,1,11] => ?
=> ? = 10 + 1
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,7,6,9,5,4,3,2,1,10] => [[1,4,10],[2],[3],[5],[6],[7],[8],[9]]
=> ? = 9 + 1
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,9,8,7,5,4,3,2,1,10] => ?
=> ? = 9 + 1
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [7,6,5,4,8,3,2,1,9] => ?
=> ? = 8 + 1
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [6,7,5,8,4,3,2,1,9] => [[1,2,4,9],[3],[5],[6],[7],[8]]
=> ? = 8 + 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [6,5,8,7,4,3,2,1,9] => [[1,3,9],[2,4],[5],[6],[7],[8]]
=> ? = 8 + 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,7,8,6,4,3,2,1,9] => ?
=> ? = 8 + 1
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [3,6,7,5,4,2,1,8] => ?
=> ? = 7 + 1
[5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,6,7,5,4,8,3,2] => ?
=> ? = 5 + 1
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,8,7,6,5,9,4,3,2] => ?
=> ? = 5 + 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,7,8,6,9,5,4,3,2] => ?
=> ? = 4 + 1
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,9,8,7,10,6,5,4,3,2] => ?
=> ? = 4 + 1
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,5,6,8,7,4,3,2] => ?
=> ? = 3 + 1
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,7,6,9,8,5,4,3,2] => ?
=> ? = 3 + 1
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,6,8,9,7,5,4,3,2] => ?
=> ? = 3 + 1
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10,9,11,8,7,6,5,4,3,2] => ?
=> ? = 3 + 1
[2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,5,9,8,7,6,4,3,2] => ?
=> ? = 2 + 1
[2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,7,10,9,8,6,5,4,3,2] => ?
=> ? = 2 + 1
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9,11,10,8,7,6,5,4,3,2] => ?
=> ? = 2 + 1
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,12,10,9,8,7,6,5,4,3,2] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 13 + 1
[12,1]
=> [1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 12 + 1
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 11 + 1
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 11 + 1
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 10 + 1
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,10,7,6,5,4,3,2,1,11] => ?
=> ? = 10 + 1
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 10 + 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [8,7,6,5,9,4,3,2,1,10] => ?
=> ? = 9 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000476
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00132: Dyck paths āswitch returns and last double riseā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
St000476: Dyck paths ā¶ ā¤Result quality: 39% āvalues known / values provided: 50%ādistinct values known / distinct values provided: 39%
Mp00132: Dyck paths āswitch returns and last double riseā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
St000476: Dyck paths ā¶ ā¤Result quality: 39% āvalues known / values provided: 50%ādistinct values known / distinct values provided: 39%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 5
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 11
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 9
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 8
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 12
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 11
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 9
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 9
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 8
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 8
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
[6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
The following 208 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000439The position of the first down step of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000839The largest opener of a set partition. St000382The first part of an integer composition. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000026The position of the first return of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a 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. St000381The largest part of an integer composition. St000808The number of up steps of the associated bargraph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence 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. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St000691The number of changes of a binary word. St001437The flex of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St000011The number of touch points (or returns) of a Dyck path. St000653The last descent of a permutation. St000676The number of odd rises of a Dyck path. St000651The maximal size of a rise in a permutation. St001814The number of partitions interlacing the given partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000019The cardinality of the support of a permutation. St000957The number of Bruhat lower covers of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000306The bounce count of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000652The maximal difference between successive positions of a permutation. St000971The smallest closer of a set partition. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000507The number of ascents of a standard tableau. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000007The number of saliances of the permutation. St000392The length of the longest run of ones in a binary word. St000069The number of maximal elements of a poset. St000067The inversion number of the alternating sign matrix. St000297The number of leading ones in a binary word. St001777The number of weak descents in an integer composition. St000925The number of topologically connected components of a set partition. St000204The number of internal nodes of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St001480The number of simple summands of the module J^2/J^3. St000006The dinv of a Dyck path. St000051The size of the left subtree of a binary tree. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. 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. St000133The "bounce" of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000443The number of long tunnels of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St001187The number of simple modules with grade at least one in the corresponding Nakayama 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. 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. St000678The number of up steps after the last double rise of a Dyck path. St000025The number of initial rises of a Dyck path. St000105The number of blocks in the set partition. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000053The number of valleys of the 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 $eA$. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St000308The height of the tree associated to a permutation. 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. St000167The number of leaves of an ordered tree. St000527The width of the poset. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000667The greatest common divisor of the parts of the partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000668The least common multiple of the parts of the partition. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000770The major index of an integer partition when read from bottom to top. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St000332The positive inversions of an alternating sign matrix. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000383The last part of an integer composition. St001462The number of factors of a standard tableaux under concatenation. St000840The number of closers smaller than the largest opener in a perfect matching. St000542The number of left-to-right-minima 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000094The depth of an ordered tree. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000305The inverse major index of a permutation. St000331The number of upper interactions of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. 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. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000083The number of left oriented leafs of a binary tree except the first one. St000061The number of nodes on the left branch of a binary tree. St000093The cardinality of a maximal independent set of vertices of a graph. 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. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000505The biggest entry in the block containing the 1. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001571The Cartan determinant of the integer partition. St001050The number of terminal closers of a set partition. St000528The height of a poset. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000823The number of unsplittable factors of the set partition. St000273The domination number of a graph. St000544The cop number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000234The number of global ascents of a permutation. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. 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. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001557The number of inversions of the second entry of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001343The dimension of the reduced incidence algebra of a poset. St000906The length of the shortest maximal chain in a poset. St000209Maximum difference of elements in cycles. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001717The largest size of an interval in a poset. St000144The pyramid weight of the Dyck path. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001589The nesting number of a perfect matching. St001152The number of pairs with even minimum in a perfect matching. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000501The size of the first part in the decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000056The decomposition (or block) number of a permutation. St000287The number of connected components of a graph. St000485The length of the longest cycle of a permutation. St000990The first ascent of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001530The depth of a Dyck path. St000080The rank of the poset. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra.
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