searching the database
Your data matches 171 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000475
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000475: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> 0
[1,1]
=> 2
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 0
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 0
[4,1,1]
=> 2
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 3
[2,2,2]
=> 0
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 0
[5,1,1]
=> 2
[4,3]
=> 0
[4,2,1]
=> 1
[4,1,1,1]
=> 3
[3,3,1]
=> 1
[3,2,2]
=> 0
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000674
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000674: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000674: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> 2
[1,1]
=> [1,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000247
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000247: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000247: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 0
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 0
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> {{1,2,5,6,7},{3,4}}
=> 0
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> 0
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> 0
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> 4
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> {{1,5},{2,7},{3},{4},{6}}
=> 3
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> {{1,7},{2},{3},{4},{5},{6}}
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 7
Description
The number of singleton blocks of a set partition.
Matching statistic: St000877
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00317: Integer partitions āodd partsā¶ Binary words
Mp00105: Binary words ācomplementā¶ Binary words
St000877: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00105: Binary words ācomplementā¶ Binary words
St000877: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> 0 => 1 => 0
[1,1]
=> 11 => 00 => 2
[3]
=> 1 => 0 => 1
[2,1]
=> 01 => 10 => 0
[1,1,1]
=> 111 => 000 => 3
[4]
=> 0 => 1 => 0
[3,1]
=> 11 => 00 => 2
[2,2]
=> 00 => 11 => 0
[2,1,1]
=> 011 => 100 => 1
[1,1,1,1]
=> 1111 => 0000 => 4
[5]
=> 1 => 0 => 1
[4,1]
=> 01 => 10 => 0
[3,2]
=> 10 => 01 => 1
[3,1,1]
=> 111 => 000 => 3
[2,2,1]
=> 001 => 110 => 0
[2,1,1,1]
=> 0111 => 1000 => 2
[1,1,1,1,1]
=> 11111 => 00000 => 5
[6]
=> 0 => 1 => 0
[5,1]
=> 11 => 00 => 2
[4,2]
=> 00 => 11 => 0
[4,1,1]
=> 011 => 100 => 1
[3,3]
=> 11 => 00 => 2
[3,2,1]
=> 101 => 010 => 1
[3,1,1,1]
=> 1111 => 0000 => 4
[2,2,2]
=> 000 => 111 => 0
[2,2,1,1]
=> 0011 => 1100 => 0
[2,1,1,1,1]
=> 01111 => 10000 => 3
[1,1,1,1,1,1]
=> 111111 => 000000 => 6
[7]
=> 1 => 0 => 1
[6,1]
=> 01 => 10 => 0
[5,2]
=> 10 => 01 => 1
[5,1,1]
=> 111 => 000 => 3
[4,3]
=> 01 => 10 => 0
[4,2,1]
=> 001 => 110 => 0
[4,1,1,1]
=> 0111 => 1000 => 2
[3,3,1]
=> 111 => 000 => 3
[3,2,2]
=> 100 => 011 => 1
[3,2,1,1]
=> 1011 => 0100 => 2
[3,1,1,1,1]
=> 11111 => 00000 => 5
[2,2,2,1]
=> 0001 => 1110 => 0
[2,2,1,1,1]
=> 00111 => 11000 => 1
[2,1,1,1,1,1]
=> 011111 => 100000 => 4
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 7
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St001107
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00327: Dyck paths āinverse Kreweras complementā¶ Dyck paths
St001107: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00327: Dyck paths āinverse Kreweras complementā¶ Dyck paths
St001107: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 0
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000326
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00135: Binary words ārotate front-to-backā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00135: Binary words ārotate front-to-backā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 001 => 3 = 2 + 1
[1,1]
=> 110 => 101 => 1 = 0 + 1
[3]
=> 1000 => 0001 => 4 = 3 + 1
[2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,1]
=> 1110 => 1101 => 1 = 0 + 1
[4]
=> 10000 => 00001 => 5 = 4 + 1
[3,1]
=> 10010 => 00101 => 3 = 2 + 1
[2,2]
=> 1100 => 1001 => 1 = 0 + 1
[2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[1,1,1,1]
=> 11110 => 11101 => 1 = 0 + 1
[5]
=> 100000 => 000001 => 6 = 5 + 1
[4,1]
=> 100010 => 000101 => 4 = 3 + 1
[3,2]
=> 10100 => 01001 => 2 = 1 + 1
[3,1,1]
=> 100110 => 001101 => 3 = 2 + 1
[2,2,1]
=> 11010 => 10101 => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => 111101 => 1 = 0 + 1
[6]
=> 1000000 => 0000001 => 7 = 6 + 1
[5,1]
=> 1000010 => 0000101 => 5 = 4 + 1
[4,2]
=> 100100 => 001001 => 3 = 2 + 1
[4,1,1]
=> 1000110 => 0001101 => 4 = 3 + 1
[3,3]
=> 11000 => 10001 => 1 = 0 + 1
[3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0011101 => 3 = 2 + 1
[2,2,2]
=> 11100 => 11001 => 1 = 0 + 1
[2,2,1,1]
=> 110110 => 101101 => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0111101 => 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => 1111101 => 1 = 0 + 1
[7]
=> 10000000 => 00000001 => 8 = 7 + 1
[6,1]
=> 10000010 => 00000101 => 6 = 5 + 1
[5,2]
=> 1000100 => 0001001 => 4 = 3 + 1
[5,1,1]
=> 10000110 => 00001101 => 5 = 4 + 1
[4,3]
=> 101000 => 010001 => 2 = 1 + 1
[4,2,1]
=> 1001010 => 0010101 => 3 = 2 + 1
[4,1,1,1]
=> 10001110 => 00011101 => 4 = 3 + 1
[3,3,1]
=> 110010 => 100101 => 1 = 0 + 1
[3,2,2]
=> 101100 => 011001 => 2 = 1 + 1
[3,2,1,1]
=> 1010110 => 0101101 => 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => 00111101 => 3 = 2 + 1
[2,2,2,1]
=> 111010 => 110101 => 1 = 0 + 1
[2,2,1,1,1]
=> 1101110 => 1011101 => 1 = 0 + 1
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111101 => 1 = 0 + 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: St000022
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00296: Dyck paths āKnuth-Krattenthalerā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000022: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00296: Dyck paths āKnuth-Krattenthalerā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000022: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,2,5,6] => 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000248
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
Mp00221: Set partitions āconjugateā¶ Set partitions
St000248: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00284: Standard tableaux ārowsā¶ Set partitions
Mp00221: Set partitions āconjugateā¶ Set partitions
St000248: Set partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> {{1,2}}
=> {{1},{2}}
=> 0
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> {{1,2}}
=> 2
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1,2,3}}
=> 3
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
[2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 4
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
[3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2},{3},{4},{5},{6}}
=> 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> {{1,3},{2},{4},{5},{6}}
=> 0
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> {{1,2,3},{4},{5},{6}}
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> {{1,4},{2},{3},{5},{6}}
=> 0
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> {{1,2,4},{3},{5},{6}}
=> 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> {{1,2,3,4},{5},{6}}
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> {{1,3,5},{2},{4},{6}}
=> 0
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> {{1,2,3,5},{4},{6}}
=> 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> {{1,2,3,4,5},{6}}
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> {{1,2,3,4,5,6}}
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> {{1,2,3,4,5,6},{7}}
=> {{1,2},{3},{4},{5},{6},{7}}
=> 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> {{1,2,3,4,5},{6,7}}
=> {{1,3},{2},{4},{5},{6},{7}}
=> 0
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> {{1,2,3},{4},{5},{6},{7}}
=> 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> {{1,4},{2},{3},{5},{6},{7}}
=> 0
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> {{1,2,4},{3},{5},{6},{7}}
=> 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> {{1,2,3,4},{5},{6},{7}}
=> 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> {{1,2,5},{3},{4},{6},{7}}
=> 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> {{1,3,5},{2},{4},{6},{7}}
=> 0
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> {{1,2,3,5},{4},{6},{7}}
=> 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> {{1,2,3,4,5},{6},{7}}
=> 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> {{1,2,4,6},{3},{5},{7}}
=> 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> {{1,2,3,4,6},{5},{7}}
=> 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> {{1,2},{3},{4},{5},{6},{7}}
=> {{1,2,3,4,5,6},{7}}
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> {{1,2,3,4,5,6,7}}
=> 7
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000297
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00095: Integer partitions āto binary wordā¶ Binary words
Mp00105: Binary words ācomplementā¶ Binary words
Mp00135: Binary words ārotate front-to-backā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00105: Binary words ācomplementā¶ Binary words
Mp00135: Binary words ārotate front-to-backā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 011 => 110 => 2
[1,1]
=> 110 => 001 => 010 => 0
[3]
=> 1000 => 0111 => 1110 => 3
[2,1]
=> 1010 => 0101 => 1010 => 1
[1,1,1]
=> 1110 => 0001 => 0010 => 0
[4]
=> 10000 => 01111 => 11110 => 4
[3,1]
=> 10010 => 01101 => 11010 => 2
[2,2]
=> 1100 => 0011 => 0110 => 0
[2,1,1]
=> 10110 => 01001 => 10010 => 1
[1,1,1,1]
=> 11110 => 00001 => 00010 => 0
[5]
=> 100000 => 011111 => 111110 => 5
[4,1]
=> 100010 => 011101 => 111010 => 3
[3,2]
=> 10100 => 01011 => 10110 => 1
[3,1,1]
=> 100110 => 011001 => 110010 => 2
[2,2,1]
=> 11010 => 00101 => 01010 => 0
[2,1,1,1]
=> 101110 => 010001 => 100010 => 1
[1,1,1,1,1]
=> 111110 => 000001 => 000010 => 0
[6]
=> 1000000 => 0111111 => 1111110 => 6
[5,1]
=> 1000010 => 0111101 => 1111010 => 4
[4,2]
=> 100100 => 011011 => 110110 => 2
[4,1,1]
=> 1000110 => 0111001 => 1110010 => 3
[3,3]
=> 11000 => 00111 => 01110 => 0
[3,2,1]
=> 101010 => 010101 => 101010 => 1
[3,1,1,1]
=> 1001110 => 0110001 => 1100010 => 2
[2,2,2]
=> 11100 => 00011 => 00110 => 0
[2,2,1,1]
=> 110110 => 001001 => 010010 => 0
[2,1,1,1,1]
=> 1011110 => 0100001 => 1000010 => 1
[1,1,1,1,1,1]
=> 1111110 => 0000001 => 0000010 => 0
[7]
=> 10000000 => 01111111 => 11111110 => 7
[6,1]
=> 10000010 => 01111101 => 11111010 => 5
[5,2]
=> 1000100 => 0111011 => 1110110 => 3
[5,1,1]
=> 10000110 => 01111001 => 11110010 => 4
[4,3]
=> 101000 => 010111 => 101110 => 1
[4,2,1]
=> 1001010 => 0110101 => 1101010 => 2
[4,1,1,1]
=> 10001110 => 01110001 => 11100010 => 3
[3,3,1]
=> 110010 => 001101 => 011010 => 0
[3,2,2]
=> 101100 => 010011 => 100110 => 1
[3,2,1,1]
=> 1010110 => 0101001 => 1010010 => 1
[3,1,1,1,1]
=> 10011110 => 01100001 => 11000010 => 2
[2,2,2,1]
=> 111010 => 000101 => 001010 => 0
[2,2,1,1,1]
=> 1101110 => 0010001 => 0100010 => 0
[2,1,1,1,1,1]
=> 10111110 => 01000001 => 10000010 => 1
[1,1,1,1,1,1,1]
=> 11111110 => 00000001 => 00000010 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000974
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00034: Dyck paths āto binary tree: up step, left tree, down step, right treeā¶ Binary trees
Mp00010: Binary trees āto ordered tree: left child = left brotherā¶ Ordered trees
St000974: Ordered trees ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00034: Dyck paths āto binary tree: up step, left tree, down step, right treeā¶ Binary trees
Mp00010: Binary trees āto ordered tree: left child = left brotherā¶ Ordered trees
St000974: Ordered trees ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> [[[]]]
=> 2
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> [[],[]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [[[[]]]]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [[[],[]]]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> [[[]],[]]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [[],[],[]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [[[[[]]]],[]]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[[]]]]]]]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [[[[[[],[]]]]]]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [[[[[[]],[]]]]]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [[[[[[]]],[]]]]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> [[[[[[]]]],[]]]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [[[[[[]]]]],[]]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [[[[[[[],[]]]]]]]
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [[[[[],[],[]]]]]
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[[[[[[]],[]]]]]]
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> [[[[],[[]],[]]]]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[[[[[[]]],[]]]]]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> [[[],[[[]]],[]]]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [[[[[[[]]]],[]]]]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[.,.],[.,[.,[.,.]]]],.]
=> [[],[[[[]]]],[]]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [[[[[[[]]]]],[]]]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [[[[[[[]]]]]],[]]
=> 0
Description
The length of the trunk of an ordered tree.
This is the length of the path from the root to the first vertex which has not exactly one child.
The following 161 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001691The number of kings in a graph. St000011The number of touch points (or returns) of a Dyck path. St000382The first part of an integer composition. St000765The number of weak records in an integer composition. St000873The aix statistic of a permutation. St000007The number of saliances of the permutation. St001459The number of zero columns in the nullspace of a graph. St000461The rix statistic of a permutation. St000148The number of odd parts of a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000895The number of ones on the main diagonal of an alternating sign matrix. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000315The number of isolated vertices of a graph. St000338The number of pixed points of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000693The modular (standard) major index of a standard tableau. St000214The number of adjacencies of a permutation. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St000894The trace of an alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St000239The number of small weak excedances. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000335The difference of lower and upper interactions. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St000010The length of the partition. St000012The area of a Dyck path. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000288The number of ones in a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St000259The diameter of a connected graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000478Another weight of a partition according to Alladi. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a 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. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001525The number of symmetric hooks on the diagonal of a partition. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001903The number of fixed points of a parking function. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. 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. 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. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001267The length of the Lyndon factorization of the binary word. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000284The Plancherel distribution on integer partitions. St000455The second largest eigenvalue of a graph if it is integral. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001060The distinguishing index of a graph. 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. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001645The pebbling number of a connected graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database ā it's very simple and we need your support!