Your data matches 23 different statistics following compositions of up to 3 maps.
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Mp00311: Plane partitions to partitionInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [3,3]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [3,3]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [3,2,1]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [3,2,1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [3,2,1]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [3,2,1]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [7]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [6,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [5,2]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [4,3]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [6,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [5,2]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [4,3]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [3,3,1]
=> 2
[[3],[3],[1]]
=> [3,3,1]
=> [3,2,2]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [3,3,1]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [3,3,1]
=> 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,2,2]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000297
Mp00311: Plane partitions to partitionInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000297: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> 1000 => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> 10000 => 1
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 10010 => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 10010 => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> 10100 => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> 10100 => 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> 100100 => 1
[[2],[2],[2]]
=> [2,2,2]
=> [3,3]
=> 11000 => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> 100100 => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [3,3]
=> 11000 => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[[3],[2],[1]]
=> [3,2,1]
=> [3,2,1]
=> 101010 => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [3,2,1]
=> 101010 => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [3,2,1]
=> 101010 => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [3,2,1]
=> 101010 => 1
[[4],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [7]
=> 10000000 => 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [6,1]
=> 10000010 => 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [5,2]
=> 1000100 => 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [4,3]
=> 101000 => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [6,1]
=> 10000010 => 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [5,2]
=> 1000100 => 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [4,3]
=> 101000 => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> 1001010 => 1
[[3],[2],[2]]
=> [3,2,2]
=> [3,3,1]
=> 110010 => 2
[[3],[3],[1]]
=> [3,3,1]
=> [3,2,2]
=> 101100 => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> 1001010 => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [3,3,1]
=> 110010 => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> 1001010 => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [3,3,1]
=> 110010 => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,2,2]
=> 101100 => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> 1001010 => 1
Description
The number of leading ones in a binary word.
Mp00311: Plane partitions to partitionInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00134: Standard tableaux descent wordBinary words
St000326: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 11111 => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 11010 => 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 01010 => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 11010 => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 01010 => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 11100 => 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 11100 => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 11100 => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 111111 => 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 111110 => 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 111010 => 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 101010 => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 111110 => 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 111010 => 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 101010 => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 111100 => 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 110100 => 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 010100 => 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 100100 => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 111100 => 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 110100 => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 010100 => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 110100 => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 010100 => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 100100 => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 111100 => 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 110100 => 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
Mp00311: Plane partitions to partitionInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => 1
Description
The first part of an integer composition.
Mp00311: Plane partitions to partitionInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00097: Binary words delta morphismInteger compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> 1110 => [3,1] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 11110 => [4,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 111110 => [5,1] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[[2],[2],[1]]
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
[[3],[1],[1]]
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1111110 => [6,1] => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
[[2],[2],[2]]
=> [2,2,2]
=> 11100 => [3,2] => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> 11100 => [3,2] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
[[3],[2],[1]]
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[[4],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> 11111110 => [7,1] => 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => [1,1,5,1] => 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => [2,1,3,1] => 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> 111010 => [3,1,1,1] => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => [1,1,5,1] => 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => [2,1,3,1] => 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> 111010 => [3,1,1,1] => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => [1,2,4,1] => 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => 1
[[3],[2],[2]]
=> [3,2,2]
=> 101100 => [1,1,2,2] => 2
[[3],[3],[1]]
=> [3,3,1]
=> 110010 => [2,2,1,1] => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => [1,2,4,1] => 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> 101100 => [1,1,2,2] => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> 101100 => [1,1,2,2] => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> 110010 => [2,2,1,1] => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => [1,2,4,1] => 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000667
Mp00311: Plane partitions to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[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],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1],[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
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[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]
=> [1,1,1,1]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Mp00311: Plane partitions to partitionInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mp00311: Plane partitions to partitionInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7]]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7]]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7]]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7]]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7]]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7]]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000996
Mp00311: Plane partitions to partitionInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000996: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => 1
Description
The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.
Mp00311: Plane partitions to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001038: Dyck paths ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[2],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000990The first ascent of a permutation. St000657The smallest part of an integer composition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000654The first descent of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000090The variation of a composition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000314The number of left-to-right-maxima of a permutation.