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Your data matches 222 different statistics following compositions of up to 3 maps.
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Matching statistic: St000075
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00033: Dyck paths āto two-row standard tableauā¶ Standard tableaux
St000075: Standard tableaux ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00033: Dyck paths āto two-row standard tableauā¶ Standard tableaux
St000075: Standard tableaux ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[5,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 2
[[3,1,1,1],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[[2,2,2],[1]]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3
Description
The orbit size of a standard tableau under promotion.
Matching statistic: St000236
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(load all 2 compositions to match this statistic)
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000236: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000236: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 2
[[3,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [2,1] => 2
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[[2,2,2],[1]]
=> [1]
=> [1,0]
=> [1] => 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 3
Description
The number of cyclical small weak excedances.
A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
Matching statistic: St000495
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000495: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000495: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[5,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[3,3,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[4,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[3,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[4,3,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
Description
The number of inversions of distance at most 2 of a permutation.
An inversion of a permutation $\pi$ is a pair $i < j$ such that $\sigma(i) > \sigma(j)$. Let $j-i$ be the distance of such an inversion. Then inversions of distance at most 1 are then exactly the descents of $\pi$, see [[St000021]]. This statistic counts the number of inversions of distance at most 2.
Matching statistic: St000519
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00269: Binary words āflag zeros to zerosā¶ Binary words
St000519: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00269: Binary words āflag zeros to zerosā¶ Binary words
St000519: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> 10 => 00 => 1
[[3,1],[1]]
=> [1]
=> 10 => 00 => 1
[[2,2],[1]]
=> [1]
=> 10 => 00 => 1
[[3,2],[2]]
=> [2]
=> 100 => 010 => 2
[[2,2,1],[1,1]]
=> [1,1]
=> 110 => 001 => 2
[[2,1,1],[1]]
=> [1]
=> 10 => 00 => 1
[[3,2,1],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[4,1],[1]]
=> [1]
=> 10 => 00 => 1
[[3,2],[1]]
=> [1]
=> 10 => 00 => 1
[[4,2],[2]]
=> [2]
=> 100 => 010 => 2
[[3,2,1],[1,1]]
=> [1,1]
=> 110 => 001 => 2
[[3,1,1],[1]]
=> [1]
=> 10 => 00 => 1
[[4,2,1],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[3,3],[2]]
=> [2]
=> 100 => 010 => 2
[[4,3],[3]]
=> [3]
=> 1000 => 0110 => 2
[[2,2,1],[1]]
=> [1]
=> 10 => 00 => 1
[[3,3,1],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[3,2,1],[2]]
=> [2]
=> 100 => 010 => 2
[[2,2,2],[1,1]]
=> [1,1]
=> 110 => 001 => 2
[[3,3,2],[2,2]]
=> [2,2]
=> 1100 => 0101 => 3
[[3,2,2],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0011 => 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 001 => 2
[[2,1,1,1],[1]]
=> [1]
=> 10 => 00 => 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[5,1],[1]]
=> [1]
=> 10 => 00 => 1
[[4,2],[1]]
=> [1]
=> 10 => 00 => 1
[[5,2],[2]]
=> [2]
=> 100 => 010 => 2
[[4,2,1],[1,1]]
=> [1,1]
=> 110 => 001 => 2
[[4,1,1],[1]]
=> [1]
=> 10 => 00 => 1
[[5,2,1],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[3,3],[1]]
=> [1]
=> 10 => 00 => 1
[[4,3],[2]]
=> [2]
=> 100 => 010 => 2
[[5,3],[3]]
=> [3]
=> 1000 => 0110 => 2
[[3,3,1],[1,1]]
=> [1,1]
=> 110 => 001 => 2
[[3,2,1],[1]]
=> [1]
=> 10 => 00 => 1
[[4,3,1],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[4,2,1],[2]]
=> [2]
=> 100 => 010 => 2
[[3,2,2],[1,1]]
=> [1,1]
=> 110 => 001 => 2
[[4,3,2],[2,2]]
=> [2,2]
=> 1100 => 0101 => 3
[[4,2,2],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0011 => 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> 110 => 001 => 2
[[3,1,1,1],[1]]
=> [1]
=> 10 => 00 => 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
[[4,4],[3]]
=> [3]
=> 1000 => 0110 => 2
[[3,3,1],[2]]
=> [2]
=> 100 => 010 => 2
[[4,3,1],[3]]
=> [3]
=> 1000 => 0110 => 2
[[2,2,2],[1]]
=> [1]
=> 10 => 00 => 1
[[3,3,2],[2,1]]
=> [2,1]
=> 1010 => 0000 => 3
Description
The largest length of a factor maximising the subword complexity.
Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal:
$$
H_w = \max\{n: p_w(n)\text{ is maximal}\}
$$
A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, [[St000294]].
Matching statistic: St000780
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
St000780: Perfect matchings ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
St000780: Perfect matchings ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[5,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[3,1,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[[2,2,2],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
Description
The size of the orbit under rotation of a perfect matching.
The number of orbits is given in [1].
Matching statistic: St000945
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
St000945: Perfect matchings ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
St000945: Perfect matchings ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[5,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 2
[[3,1,1,1],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 2
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
[[2,2,2],[1]]
=> [1]
=> [1,0]
=> [(1,2)]
=> 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 3
Description
The number of matchings in the dihedral orbit of a perfect matching.
The dihedral orbit is induced by the dihedral symmetry of the underlying circular arrangement of points. In other words, this is the number of matchings that can be obtained by rotating or reflecting the given perfect matching.
Matching statistic: St001183
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St001183: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St001183: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
Description
The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001258
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St001258: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St001258: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,3],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[2,2,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.
For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
Matching statistic: St001486
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00295: Standard tableaux āvalley compositionā¶ Integer compositions
St001486: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00295: Standard tableaux āvalley compositionā¶ Integer compositions
St001486: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[2,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2],[2]]
=> [2]
=> [[1,2]]
=> [2] => 2
[[2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2] => 2
[[2,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[4,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,2],[2]]
=> [2]
=> [[1,2]]
=> [2] => 2
[[3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2] => 2
[[3,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[3,3],[2]]
=> [2]
=> [[1,2]]
=> [2] => 2
[[4,3],[3]]
=> [3]
=> [[1,2,3]]
=> [3] => 2
[[2,2,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[3,2,1],[2]]
=> [2]
=> [[1,2]]
=> [2] => 2
[[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2] => 2
[[3,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,1] => 3
[[3,2,2],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3] => 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2] => 2
[[2,1,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[5,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[5,2],[2]]
=> [2]
=> [[1,2]]
=> [2] => 2
[[4,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2] => 2
[[4,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[5,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[3,3],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,3],[2]]
=> [2]
=> [[1,2]]
=> [2] => 2
[[5,3],[3]]
=> [3]
=> [[1,2,3]]
=> [3] => 2
[[3,3,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2] => 2
[[3,2,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,3,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[4,2,1],[2]]
=> [2]
=> [[1,2]]
=> [2] => 2
[[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2] => 2
[[4,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,1] => 3
[[4,2,2],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3] => 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2] => 2
[[3,1,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
[[4,4],[3]]
=> [3]
=> [[1,2,3]]
=> [3] => 2
[[3,3,1],[2]]
=> [2]
=> [[1,2]]
=> [2] => 2
[[4,3,1],[3]]
=> [3]
=> [[1,2,3]]
=> [3] => 2
[[2,2,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,3,2],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [2,1] => 3
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St001526
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions āinner shapeā¶ Integer partitions
Mp00312: Integer partitions āGlaisher-Franklinā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001526: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00312: Integer partitions āGlaisher-Franklinā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001526: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[2,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[3,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[[2,2,1],[1,1]]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[4,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[3,2],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,2,1],[1,1]]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[3,3],[2]]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[[2,2,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[3,2,1],[2]]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[5,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[4,2],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[[4,2,1],[1,1]]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[3,3],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[4,3],[2]]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[[5,3],[3]]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[[3,3,1],[1,1]]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[[3,2,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[4,2,1],[2]]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[[4,2,2],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[3,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[[3,1,1,1],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[4,4],[3]]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[[3,3,1],[2]]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 2
[[4,3,1],[3]]
=> [3]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[[2,2,2],[1]]
=> [1]
=> [1]
=> [1,0]
=> 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
The following 212 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000845The maximal number of elements covered by an element in a poset. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001424The number of distinct squares in a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000100The number of linear extensions of a poset. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000988The orbit size of a permutation under Foata's bijection. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001313The number of Dyck paths above the lattice path given by a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000628The balance of a binary word. St000682The Grundy value of Welter's game on a binary word. St000747A variant of the major index of a set partition. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001388The number of non-attacking neighbors of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001731The factorization defect of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000929The constant term of the character polynomial of an integer partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000937The number of positive values of the symmetric group character corresponding to the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000567The sum of the products of all pairs of parts. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000997The even-odd crank of an integer partition. St000438The position of the last up step in a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001808The box weight or horizontal decoration of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St001204Call 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St000474Dyson's crank of a partition.
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