Your data matches 54 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000729
Mapping
St000729: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => 1
{{1},{2}}
 => 2
{{1,2,3}}
 => 1
{{1,2},{3}}
 => 1
{{1,3},{2}}
 => 2
{{1},{2,3}}
 => 1
{{1},{2},{3}}
 => 3
{{1,2,3,4}}
 => 1
{{1,2,3},{4}}
 => 1
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 1
{{1,2},{3},{4}}
 => 1
{{1,3,4},{2}}
 => 1
{{1,3},{2,4}}
 => 2
{{1,3},{2},{4}}
 => 2
{{1,4},{2,3}}
 => 1
{{1},{2,3,4}}
 => 1
{{1},{2,3},{4}}
 => 1
{{1,4},{2},{3}}
 => 3
{{1},{2,4},{3}}
 => 2
{{1},{2},{3,4}}
 => 1
{{1},{2},{3},{4}}
 => 4
{{1,2,3,4,5}}
 => 1
{{1,2,3,4},{5}}
 => 1
{{1,2,3,5},{4}}
 => 1
{{1,2,3},{4,5}}
 => 1
{{1,2,3},{4},{5}}
 => 1
{{1,2,4,5},{3}}
 => 1
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 1
{{1,2},{3,4,5}}
 => 1
{{1,2},{3,4},{5}}
 => 1
{{1,2,5},{3},{4}}
 => 1
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 1
{{1,2},{3},{4},{5}}
 => 1
{{1,3,4,5},{2}}
 => 1
{{1,3,4},{2,5}}
 => 1
{{1,3,4},{2},{5}}
 => 1
{{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 2
{{1,3,5},{2},{4}}
 => 2
{{1,3},{2,5},{4}}
 => 2
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 2
{{1,4,5},{2,3}}
 => 1
{{1,4},{2,3,5}}
 => 1
{{1,4},{2,3},{5}}
 => 1
Description
The minimal arc length of a set partition. The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the minimal arc length is the size of the ground set (as the minimum of the empty set in the universe of arcs of length less than the size of the ground set).
Matching statistic: St000297
(load all 7 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => 10 => 1
{{1},{2}}
 => [1,1] => 11 => 2
{{1,2,3}}
 => [3] => 100 => 1
{{1,2},{3}}
 => [2,1] => 101 => 1
{{1,3},{2}}
 => [2,1] => 101 => 1
{{1},{2,3}}
 => [1,2] => 110 => 2
{{1},{2},{3}}
 => [1,1,1] => 111 => 3
{{1,2,3,4}}
 => [4] => 1000 => 1
{{1,2,3},{4}}
 => [3,1] => 1001 => 1
{{1,2,4},{3}}
 => [3,1] => 1001 => 1
{{1,2},{3,4}}
 => [2,2] => 1010 => 1
{{1,2},{3},{4}}
 => [2,1,1] => 1011 => 1
{{1,3,4},{2}}
 => [3,1] => 1001 => 1
{{1,3},{2,4}}
 => [2,2] => 1010 => 1
{{1,3},{2},{4}}
 => [2,1,1] => 1011 => 1
{{1,4},{2,3}}
 => [2,2] => 1010 => 1
{{1},{2,3,4}}
 => [1,3] => 1100 => 2
{{1},{2,3},{4}}
 => [1,2,1] => 1101 => 2
{{1,4},{2},{3}}
 => [2,1,1] => 1011 => 1
{{1},{2,4},{3}}
 => [1,2,1] => 1101 => 2
{{1},{2},{3,4}}
 => [1,1,2] => 1110 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => 1111 => 4
{{1,2,3,4,5}}
 => [5] => 10000 => 1
{{1,2,3,4},{5}}
 => [4,1] => 10001 => 1
{{1,2,3,5},{4}}
 => [4,1] => 10001 => 1
{{1,2,3},{4,5}}
 => [3,2] => 10010 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => 10011 => 1
{{1,2,4,5},{3}}
 => [4,1] => 10001 => 1
{{1,2,4},{3,5}}
 => [3,2] => 10010 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => 10011 => 1
{{1,2,5},{3,4}}
 => [3,2] => 10010 => 1
{{1,2},{3,4,5}}
 => [2,3] => 10100 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => 10101 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => 10011 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => 10101 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => 10110 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => 10111 => 1
{{1,3,4,5},{2}}
 => [4,1] => 10001 => 1
{{1,3,4},{2,5}}
 => [3,2] => 10010 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => 10011 => 1
{{1,3,5},{2,4}}
 => [3,2] => 10010 => 1
{{1,3},{2,4,5}}
 => [2,3] => 10100 => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => 10101 => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => 10011 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => 10101 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => 10110 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => 10111 => 1
{{1,4,5},{2,3}}
 => [3,2] => 10010 => 1
{{1,4},{2,3,5}}
 => [2,3] => 10100 => 1
{{1,4},{2,3},{5}}
 => [2,2,1] => 10101 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 14 compositions to match this statistic)
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [[1,2]]
 => 0 => 2
{{1},{2}}
 => [[1],[2]]
 => 1 => 1
{{1,2,3}}
 => [[1,2,3]]
 => 00 => 3
{{1,2},{3}}
 => [[1,2],[3]]
 => 01 => 2
{{1,3},{2}}
 => [[1,3],[2]]
 => 10 => 1
{{1},{2,3}}
 => [[1,3],[2]]
 => 10 => 1
{{1},{2},{3}}
 => [[1],[2],[3]]
 => 11 => 1
{{1,2,3,4}}
 => [[1,2,3,4]]
 => 000 => 4
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 001 => 3
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 010 => 2
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 010 => 2
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 011 => 2
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 100 => 1
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => 101 => 1
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 101 => 1
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => 101 => 1
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 100 => 1
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 101 => 1
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 110 => 1
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 110 => 1
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 110 => 1
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 111 => 1
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => 0000 => 5
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => 0001 => 4
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => 0010 => 3
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => 0010 => 3
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => 0011 => 3
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => 0100 => 2
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => 0101 => 2
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => 0101 => 2
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => 0100 => 2
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => 0100 => 2
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => 0101 => 2
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => 0110 => 2
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => 0110 => 2
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => 0110 => 2
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => 0111 => 2
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => 1000 => 1
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => 1001 => 1
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => 1001 => 1
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => 1010 => 1
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => 1010 => 1
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => 1011 => 1
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => 1010 => 1
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => 1010 => 1
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => 1010 => 1
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => 1011 => 1
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => 1010 => 1
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => 1010 => 1
{{1,4},{2,3},{5}}
 => [[1,3],[2,4],[5]]
 => 1011 => 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
(load all 5 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1] => 1
{{1},{2}}
 => [1,1] => [2] => 2
{{1,2,3}}
 => [3] => [1,1,1] => 1
{{1,2},{3}}
 => [2,1] => [1,2] => 1
{{1,3},{2}}
 => [2,1] => [1,2] => 1
{{1},{2,3}}
 => [1,2] => [2,1] => 2
{{1},{2},{3}}
 => [1,1,1] => [3] => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,2] => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,2] => 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,3] => 1
{{1,3,4},{2}}
 => [3,1] => [1,1,2] => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,3] => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => 1
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => 2
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,3] => 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,2] => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,2] => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,2] => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,4] => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 7 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1] => 1
{{1},{2}}
 => [1,1] => [2] => 2
{{1,2,3}}
 => [3] => [1,1,1] => 1
{{1,2},{3}}
 => [2,1] => [2,1] => 1
{{1,3},{2}}
 => [2,1] => [2,1] => 1
{{1},{2,3}}
 => [1,2] => [1,2] => 2
{{1},{2},{3}}
 => [1,1,1] => [3] => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
 => [3,1] => [2,1,1] => 1
{{1,2,4},{3}}
 => [3,1] => [2,1,1] => 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => 1
{{1,2},{3},{4}}
 => [2,1,1] => [3,1] => 1
{{1,3,4},{2}}
 => [3,1] => [2,1,1] => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => 1
{{1,3},{2},{4}}
 => [2,1,1] => [3,1] => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => 1
{{1},{2,3,4}}
 => [1,3] => [1,1,2] => 2
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 2
{{1,4},{2},{3}}
 => [2,1,1] => [3,1] => 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
 => [1,1,2] => [1,3] => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,3,5},{4}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,2,1,1] => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,2,4,5},{3}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,2,1,1] => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,2,1,1] => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,2,1] => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [3,1,1] => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [2,2,1] => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [4,1] => 1
{{1,3,4,5},{2}}
 => [4,1] => [2,1,1,1] => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,2,1,1] => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,2,1,1] => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1,2,1] => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [3,1,1] => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [2,2,1] => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [4,1] => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,2,1,1] => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,2,1] => 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [2,2,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000745
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [[1,2]]
 => [[1],[2]]
 => 2
{{1},{2}}
 => [[1],[2]]
 => [[1,2]]
 => 1
{{1,2,3}}
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
{{1,2},{3}}
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
{{1,3},{2}}
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
{{1},{2,3}}
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 2
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 1
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 1
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 1
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 1
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 4
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 3
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => 2
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 2
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => 2
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 2
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => 2
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => 2
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => 2
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 2
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 1
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => 1
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => 1
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 1
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 1
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => 1
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 1
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => 1
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 1
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 1
{{1,4},{2,3},{5}}
 => [[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000025
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1] => [1,0,1,0]
 => 1
{{1},{2}}
 => [1,1] => [2] => [1,1,0,0]
 => 2
{{1,2,3}}
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
{{1,2},{3}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1,3},{2}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
{{1},{2},{3}}
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,3,4},{2}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1] => [1,0,1,0]
 => 1
{{1},{2}}
 => [1,1] => [2] => [1,1,0,0]
 => 2
{{1,2,3}}
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
{{1,2},{3}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1,3},{2}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
{{1},{2},{3}}
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,3,4},{2}}
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [2] => [1,1,0,0]
 => 1
{{1},{2}}
 => [1,1] => [1,1] => [1,0,1,0]
 => 2
{{1,2,3}}
 => [3] => [3] => [1,1,1,0,0,0]
 => 1
{{1,2},{3}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1,3},{2}}
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2
{{1},{2},{3}}
 => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
{{1,2,3,4}}
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,2,4},{3}}
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,4}}
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,3,4},{2}}
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4}}
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1,4},{2,3}}
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
{{1},{2,3,4}}
 => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
{{1},{2,3},{4}}
 => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
{{1},{2},{3,4}}
 => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,3,4,5},{2}}
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4,5}}
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,4,5},{2,3}}
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000759
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2] => [1,1,0,0]
 => []
 => 1
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => [1]
 => 2
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => []
 => 1
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 2
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 2
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1
{{1,4,5},{2,3}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000439The position of the first down step of a Dyck path. St000363The number of minimal vertex covers of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000501The size of the first part in the decomposition of a permutation. St000260The radius of a connected graph. St000234The number of global ascents of a permutation. St000617The number of global maxima of a Dyck path. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000546The number of global descents of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000989The number of final rises of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. 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. St000054The first entry of the permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function.