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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000326
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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%
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
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(load all 7 compositions to match this statistic)
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [[1,2]]
=> [2] => 2
{{1},{2}}
=> [[1],[2]]
=> [1,1] => 1
{{1,2,3}}
=> [[1,2,3]]
=> [3] => 3
{{1,2},{3}}
=> [[1,2],[3]]
=> [2,1] => 2
{{1,3},{2}}
=> [[1,3],[2]]
=> [1,2] => 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [1,2] => 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [1,1,1] => 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [4] => 4
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [3,1] => 3
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [2,2] => 2
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [1,2,1] => 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [1,2,1] => 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [1,2,1] => 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [1,2,1] => 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [5] => 5
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [4,1] => 4
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [3,2] => 3
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [3,2] => 3
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 3
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [2,3] => 2
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [2,2,1] => 2
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [2,2,1] => 2
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [2,1,2] => 2
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [2,1,2] => 2
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [2,1,2] => 2
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [1,3,1] => 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [1,3,1] => 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [1,2,2] => 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000745
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(load all 9 compositions to match this statistic)
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%
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: St000297
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(load all 3 compositions to match this statistic)
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [[1,2]]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
{{1},{2}}
=> [[1],[2]]
=> [[1,2]]
=> 0 => 0 = 1 - 1
{{1,2,3}}
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 11 => 2 = 3 - 1
{{1,2},{3}}
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 10 => 1 = 2 - 1
{{1,3},{2}}
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 01 => 0 = 1 - 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 01 => 0 = 1 - 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 00 => 0 = 1 - 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 111 => 3 = 4 - 1
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 110 => 2 = 3 - 1
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 101 => 1 = 2 - 1
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 101 => 1 = 2 - 1
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 100 => 1 = 2 - 1
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 011 => 0 = 1 - 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 010 => 0 = 1 - 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 010 => 0 = 1 - 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 010 => 0 = 1 - 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 011 => 0 = 1 - 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 010 => 0 = 1 - 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 0 = 1 - 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 0 = 1 - 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 000 => 0 = 1 - 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 4 = 5 - 1
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 4 - 1
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 1101 => 2 = 3 - 1
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 1101 => 2 = 3 - 1
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 3 - 1
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 1011 => 1 = 2 - 1
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 1010 => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 1010 => 1 = 2 - 1
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 1011 => 1 = 2 - 1
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 1011 => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1010 => 1 = 2 - 1
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 1001 => 1 = 2 - 1
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 1001 => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 1001 => 1 = 2 - 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1000 => 1 = 2 - 1
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 0110 => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0 = 1 - 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 0100 => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 0101 => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 0101 => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 0101 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 0100 => 0 = 1 - 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 0100 => 0 = 1 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000383
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [[1,2]]
=> [2] => [2] => 2
{{1},{2}}
=> [[1],[2]]
=> [1,1] => [1,1] => 1
{{1,2,3}}
=> [[1,2,3]]
=> [3] => [3] => 3
{{1,2},{3}}
=> [[1,2],[3]]
=> [2,1] => [1,2] => 2
{{1,3},{2}}
=> [[1,3],[2]]
=> [1,2] => [2,1] => 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [1,2] => [2,1] => 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [1,1,1] => [1,1,1] => 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [4] => [4] => 4
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [3,1] => [1,3] => 3
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [2,2] => [2,2] => 2
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [2,2] => [2,2] => 2
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,1,2] => 2
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [1,3] => [3,1] => 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [1,2,1] => [1,2,1] => 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [1,2,1] => [1,2,1] => 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [1,2,1] => [1,2,1] => 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [1,3] => [3,1] => 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [1,2,1] => [1,2,1] => 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [5] => [5] => 5
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [4,1] => [1,4] => 4
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [3,2] => [2,3] => 3
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [3,2] => [2,3] => 3
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,3] => 3
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [2,3] => [3,2] => 2
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [2,2,1] => [1,2,2] => 2
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [2,2,1] => [1,2,2] => 2
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [2,3] => [3,2] => 2
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [2,3] => [3,2] => 2
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [1,2,2] => 2
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [2,1,2] => [2,1,2] => 2
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [2,1,2] => [2,1,2] => 2
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [2,1,2] => [2,1,2] => 2
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,1,2] => 2
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [1,4] => [4,1] => 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [1,3,1] => [1,3,1] => 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [1,3,1] => [1,3,1] => 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,1,2,1] => 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [1,2,2] => [2,2,1] => 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [2,2,1] => 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [2,2,1] => 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,1,2,1] => 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,1,2,1] => 1
{{1},{2,4,8},{3},{5,7},{6}}
=> [[1,4,8],[2,7],[3],[5],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,2},{3,6,7,8},{4},{5}}
=> [[1,2,7,8],[3,6],[4],[5]]
=> [2,1,1,4] => [4,1,1,2] => ? = 2
{{1,6,7,8},{2},{3},{4,5}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,4,5,6},{2},{3},{7,8}}
=> [[1,4,5,6],[2,8],[3],[7]]
=> [1,1,4,2] => [2,4,1,1] => ? = 1
{{1,4,7,8},{2},{3},{5,6}}
=> [[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => [4,2,1,1] => ? = 1
{{1,5,8},{2},{3,4},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,6,7,8},{2},{3,5},{4}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,7,8},{2},{3,6},{4,5}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,4},{2,3},{5,7,8},{6}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 1
{{1,4,8},{2,3},{5},{6,7}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 1
{{1,4,8},{2,3},{5,7},{6}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 1
{{1,5,8},{2,4},{3},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,6,7,8},{2,5},{3},{4}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,8},{2,5,6,7},{3},{4}}
=> [[1,5,6,7],[2,8],[3],[4]]
=> [1,1,1,4,1] => [1,4,1,1,1] => ? = 1
{{1,7,8},{2,6},{3},{4,5}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,7,8},{2,6},{3,5},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,5,7,8},{2,6},{3},{4}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,4,8},{2,5},{3,7},{6}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,5},{2,4,8},{3},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,4},{2,5,8},{3},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,4},{2},{3,5,8},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,5},{2},{3,4,8},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1},{2,5},{3,4,8},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1},{2,4},{3,5,8},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,7,8},{2,5},{3,6},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,5,7,8},{2},{3,6},{4}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,5,8},{2,7},{3,6},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,4,8},{2,5},{3},{6,7}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,6,8},{2,7},{3,5},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,6,8},{2,7},{3},{4,5}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,7},{2,5},{3,4,8},{6}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,5},{2,4,8},{3,7},{6}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,5},{2,7},{3,4,8},{6}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,4,8},{2,7},{3,5},{6}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,7},{2,4,8},{3,5},{6}}
=> [[1,4,8],[2,5],[3,7],[6]]
=> [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
{{1,3},{2,4,8},{5,7},{6}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 1
{{1},{2,5,8},{3,7},{4,6}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1},{2,5,8},{3,6},{4,7}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1},{2,6,8},{3,5},{4,7}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,6},{2,7},{3,5,8},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,7},{2,6,8},{3,5},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,7},{2,6},{3,5,8},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
{{1,4},{2,3,8},{5},{6,7}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 1
{{1,4},{2,3,8},{5,7},{6}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [3,1,1,2,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000729
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000729: Set partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00215: Set partitions —Wachs-White⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000729: Set partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 2
{{1},{2}}
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 1
{{1,2,3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2
{{1,3},{2}}
=> {{1},{2,3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 3
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 2
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> {{1,3},{2,4}}
=> 2
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> 1
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 1
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> {{1,3,4},{2}}
=> 1
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> {{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}}
=> {{1},{2},{3},{4},{5}}
=> 5
{{1,2,3,4},{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},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 3
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 3
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> {{1,4},{2,5},{3}}
=> 3
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 2
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> {{1},{2,4,5},{3}}
=> {{1},{2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> {{1,3},{2,5},{4}}
=> 2
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 2
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 2
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> {{1,5},{2,4},{3}}
=> 2
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1,3},{2,4},{5}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> {{1,4},{2},{3,5}}
=> 2
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> {{1,3,5},{2},{4}}
=> 2
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> {{1,3,5},{2,4}}
=> 2
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> {{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> {{1,2,4},{3,5}}
=> 1
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> {{1,4},{2,5},{3}}
=> {{1},{2,5},{3,4}}
=> 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> 1
{{1,2},{3,4},{5,6},{7,8}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> {{1},{2,4},{3,6},{5,8},{7}}
=> {{1},{2,4,8},{3,6},{5},{7}}
=> ? = 2
{{1,3},{2,4},{5,6},{7,8}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> {{1,3,4},{2,6},{5,8},{7}}
=> {{1},{2,3,4,6},{5,8},{7}}
=> ? = 1
{{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,5},{2},{4},{6,8},{7}}
=> {{1,3},{2},{4,6},{5,8},{7}}
=> {{1},{2,3,6},{4,8},{5},{7}}
=> ? = 1
{{1,5},{2,3},{4,6},{7,8}}
=> {{1,3,5},{2},{4,6,8},{7}}
=> {{1,3},{2,5,6},{4,8},{7}}
=> {{1},{2,3,5,8},{4,6},{7}}
=> ? = 1
{{1,6},{2,3},{4,5},{7,8}}
=> {{1,3},{2},{4,6,8},{5},{7}}
=> {{1,3},{2,5},{4},{6,8},{7}}
=> {{1},{2,3,5},{4,8},{6},{7}}
=> ? = 1
{{1,7},{2,3},{4,5},{6,8}}
=> {{1,3},{2,4,6,8},{5},{7}}
=> {{1,3},{2,5},{4,7,8},{6}}
=> {{1},{2,3,5,8},{4,7},{6}}
=> ? = 1
{{1,8},{2,3},{4,5},{6,7}}
=> {{1},{2,4,6,8},{3},{5},{7}}
=> {{1,3},{2,5},{4,7},{6},{8}}
=> {{1},{2,3,5},{4,7},{6},{8}}
=> ? = 1
{{1,8},{2,4},{3,5},{6,7}}
=> {{1},{2,4,6,8},{3},{5,7}}
=> {{1,3,4,5},{2,7},{6},{8}}
=> {{1},{2,3,4,5,7},{6},{8}}
=> ? = 1
{{1,7},{2,4},{3,5},{6,8}}
=> {{1,3},{2,8},{4,6},{5,7}}
=> {{1,7},{2,4,5},{3,8},{6}}
=> {{1},{2,4,8},{3,5},{6,7}}
=> ? = 1
{{1,6},{2,4},{3,5},{7,8}}
=> {{1,3},{2},{4,6,8},{5,7}}
=> {{1,3,4,5},{2},{6,8},{7}}
=> {{1},{2,3,4,5,8},{6},{7}}
=> ? = 1
{{1,5},{2,4},{3,6},{7,8}}
=> {{1,3,6},{2},{4,8},{5,7}}
=> {{1,5,6},{2,4},{3,8},{7}}
=> {{1},{2,4,5,6},{3,8},{7}}
=> ? = 1
{{1,4},{2,5},{3,6},{7,8}}
=> {{1,3,6},{2},{4,7},{5,8}}
=> {{1,4,5,6},{2},{3,8},{7}}
=> {{1},{2,8},{3,4,5,6},{7}}
=> ? = 1
{{1,3},{2,5},{4,6},{7,8}}
=> {{1,3,5},{2},{4,7},{6,8}}
=> {{1,3,5},{2,6},{4,8},{7}}
=> {{1},{2,3,6},{4,5,8},{7}}
=> ? = 1
{{1,2},{3,5},{4,6},{7,8}}
=> {{1,3,5,7},{2},{4,6},{8}}
=> {{1},{2,4,5,6},{3,8},{7}}
=> {{1},{2,4,6},{3,5,8},{7}}
=> ? = 2
{{1,2},{3,6},{4,5},{7,8}}
=> {{1,3,7},{2},{4,6},{5},{8}}
=> {{1},{2,6},{3,5},{4,8},{7}}
=> {{1},{2,5},{3,8},{4,6},{7}}
=> ? = 2
{{1,3},{2,6},{4,5},{7,8}}
=> {{1,3,7},{2},{4,6,8},{5}}
=> {{1,3,6},{2,5},{4,8},{7}}
=> {{1},{2,3,5,6},{4,8},{7}}
=> ? = 1
{{1,4},{2,6},{3,5},{7,8}}
=> {{1,3,7},{2},{4,6},{5,8}}
=> {{1,4,6},{2,5},{3,8},{7}}
=> {{1},{2,5,6},{3,4,8},{7}}
=> ? = 1
{{1,5},{2,6},{3,4},{7,8}}
=> {{1,3,5,7},{2},{4,8},{6}}
=> {{1,5},{2,4,6},{3,8},{7}}
=> {{1},{2,4,5,8},{3,6},{7}}
=> ? = 1
{{1,6},{2,5},{3,4},{7,8}}
=> {{1,3},{2},{4,8},{5,7},{6}}
=> {{1,5},{2,4},{3},{6,8},{7}}
=> {{1},{2,4,5},{3,8},{6},{7}}
=> ? = 1
{{1,7},{2,5},{3,4},{6,8}}
=> {{1,3},{2,4,8},{5,7},{6}}
=> {{1,5},{2,4},{3,7,8},{6}}
=> {{1},{2,4,5,8},{3,7},{6}}
=> ? = 1
{{1,8},{2,5},{3,4},{6,7}}
=> {{1},{2,4,8},{3},{5,7},{6}}
=> {{1,5},{2,4},{3,7},{6},{8}}
=> {{1},{2,4,5},{3,7},{6},{8}}
=> ? = 1
{{1,8},{2,6},{3,4},{5,7}}
=> {{1},{2,4,8},{3,5,7},{6}}
=> {{1,5},{2,4,6,7},{3},{8}}
=> {{1},{2,4,5,7},{3,6},{8}}
=> ? = 1
{{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2,8},{3,5,7},{6}}
=> {{1,7},{2,4,8},{3,6},{5}}
=> {{1},{2,4},{3,6,7},{5,8}}
=> ? = 1
{{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2,5,7},{3,8},{6}}
=> {{1,6},{2,4,7,8},{3},{5}}
=> {{1},{2,4,7},{3},{5,6,8}}
=> ? = 1
{{1,5},{2,7},{3,4},{6,8}}
=> {{1,3},{2,5,7},{4,8},{6}}
=> {{1,5},{2,4,7},{3,8},{6}}
=> {{1},{2,4,5,7},{3,8},{6}}
=> ? = 1
{{1,4},{2,7},{3,5},{6,8}}
=> {{1,3},{2,7},{4,6},{5,8}}
=> {{1,4,7},{2,5},{3,8},{6}}
=> {{1},{2,5},{3,4,8},{6,7}}
=> ? = 1
{{1,3},{2,7},{4,5},{6,8}}
=> {{1,3},{2,7},{4,6,8},{5}}
=> {{1,3,7},{2,5},{4,8},{6}}
=> {{1},{2,3,5},{4,8},{6,7}}
=> ? = 1
{{1,2},{3,7},{4,5},{6,8}}
=> {{1,3,7},{2,4,6},{5},{8}}
=> {{1},{2,6},{3,5,7,8},{4}}
=> {{1},{2,5,8},{3,7},{4,6}}
=> ? = 2
{{1,2},{3,8},{4,5},{6,7}}
=> {{1,7},{2,4,6},{3},{5},{8}}
=> {{1},{2,8},{3,5},{4,7},{6}}
=> {{1},{2,5},{3,7},{4},{6,8}}
=> ? = 2
{{1,3},{2,8},{4,5},{6,7}}
=> {{1,7},{2,4,6,8},{3},{5}}
=> {{1,3,8},{2,5},{4,7},{6}}
=> {{1},{2,3,5},{4,7,8},{6}}
=> ? = 1
{{1,4},{2,8},{3,5},{6,7}}
=> {{1,7},{2,4,6},{3},{5,8}}
=> {{1,4,8},{2,5},{3,7},{6}}
=> {{1},{2,5},{3,4,7,8},{6}}
=> ? = 1
{{1,5},{2,8},{3,4},{6,7}}
=> {{1,5,7},{2,4,8},{3},{6}}
=> {{1,5},{2,4,8},{3,7},{6}}
=> {{1},{2,4,5},{3,7},{6,8}}
=> ? = 1
{{1,6},{2,8},{3,4},{5,7}}
=> {{1,5,7},{2,4},{3,8},{6}}
=> {{1,6},{2,4,8},{3,7},{5}}
=> {{1},{2,4},{3,7},{5,6,8}}
=> ? = 1
{{1,7},{2,8},{3,4},{5,6}}
=> {{1,3,5,7},{2,8},{4},{6}}
=> {{1,7},{2,4},{3,6,8},{5}}
=> {{1},{2,4,8},{3,6,7},{5}}
=> ? = 1
{{1,8},{2,7},{3,4},{5,6}}
=> {{1},{2,8},{3,5,7},{4},{6}}
=> {{1,7},{2,4},{3,6},{5},{8}}
=> {{1},{2,4},{3,6,7},{5},{8}}
=> ? = 1
{{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,8},{3,5,7},{4,6}}
=> {{1,7},{2,4,5,6},{3},{8}}
=> {{1},{2,4,6,7},{3,5},{8}}
=> ? = 1
{{1,7},{2,8},{3,5},{4,6}}
=> {{1,7},{2,8},{3,5},{4,6}}
=> {{1,7,8},{2},{3,5,6},{4}}
=> {{1},{2,5},{3,6,7,8},{4}}
=> ? = 1
{{1,6},{2,8},{3,5},{4,7}}
=> {{1,7},{2,5},{3,8},{4,6}}
=> {{1,6,8},{2,7},{3,5},{4}}
=> {{1},{2,5,6},{3},{4,7,8}}
=> ? = 1
{{1,5},{2,8},{3,6},{4,7}}
=> {{1,7},{2,5},{3,6},{4,8}}
=> {{1,5,8},{2,6,7},{3},{4}}
=> {{1},{2},{3,6},{4,5,7,8}}
=> ? = 1
{{1,4},{2,8},{3,6},{5,7}}
=> {{1,7},{2,4},{3,6},{5,8}}
=> {{1,4,8},{2,6},{3,7},{5}}
=> {{1},{2,7,8},{3,4,6},{5}}
=> ? = 1
{{1,3},{2,8},{4,6},{5,7}}
=> {{1,7},{2,4},{3,5},{6,8}}
=> {{1,3,8},{2},{4,6,7},{5}}
=> {{1},{2,3,6},{4,7,8},{5}}
=> ? = 1
{{1,2},{3,8},{4,6},{5,7}}
=> {{1,7},{2,4,6},{3,5},{8}}
=> {{1},{2,8},{3,5,6,7},{4}}
=> {{1},{2,5},{3,6,8},{4,7}}
=> ? = 2
{{1,2},{3,7},{4,6},{5,8}}
=> {{1,4,7},{2,6},{3,5},{8}}
=> {{1},{2,5,7,8},{3,6},{4}}
=> {{1},{2,6,8},{3,5,7},{4}}
=> ? = 2
{{1,3},{2,7},{4,6},{5,8}}
=> {{1,4},{2,7},{3,5},{6,8}}
=> {{1,3,7},{2,8},{4,6},{5}}
=> {{1},{2,3,6,7},{4},{5,8}}
=> ? = 1
{{1,4},{2,7},{3,6},{5,8}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> {{1,4,7},{2,6,8},{3},{5}}
=> {{1},{2},{3,4,6,7},{5,8}}
=> ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> {{1,5},{2,7},{3,6},{4,8}}
=> {{1,5,7,8},{2,6},{3},{4}}
=> {{1},{2},{3,6,7,8},{4,5}}
=> ? = 1
{{1,6},{2,7},{3,5},{4,8}}
=> {{1,5},{2,7},{3,8},{4,6}}
=> {{1,6,7,8},{2},{3,5},{4}}
=> {{1},{2,5,6,7,8},{3},{4}}
=> ? = 1
{{1,7},{2,6},{3,5},{4,8}}
=> {{1,5},{2,8},{3,7},{4,6}}
=> {{1,7,8},{2,6},{3,5},{4}}
=> {{1},{2,5},{3},{4,6,7,8}}
=> ? = 1
{{1,8},{2,6},{3,5},{4,7}}
=> {{1},{2,5,8},{3,7},{4,6}}
=> {{1,4,6,7},{2,5},{3},{8}}
=> {{1},{2,5,6,7},{3,4},{8}}
=> ? = 1
{{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2,5,8},{3,6},{4,7}}
=> {{1,4,5,6,7},{2},{3},{8}}
=> {{1},{2},{3,4,5,6,7},{8}}
=> ? = 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: St000260
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00215: Set partitions —Wachs-White⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 42%●distinct values known / distinct values provided: 14%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 42%●distinct values known / distinct values provided: 14%
Values
{{1,2}}
=> {{1,2}}
=> [2] => ([],2)
=> ? = 2
{{1},{2}}
=> {{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 1
{{1,2,3}}
=> {{1,2,3}}
=> [3] => ([],3)
=> ? = 3
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> ? = 2
{{1,3},{2}}
=> {{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => ([],4)
=> ? = 4
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> ? = 3
{{1,2,4},{3}}
=> {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1,3},{2,4}}
=> {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5] => ([],5)
=> ? = 5
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> ? = 4
{{1,2,3,5},{4}}
=> {{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3
{{1,2,4,5},{3}}
=> {{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
{{1,2,4},{3,5}}
=> {{1,3,4},{2,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
{{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2,5},{3,4}}
=> {{1,4},{2,3,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2,5},{3},{4}}
=> {{1,4},{2},{3,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1,3,4},{2,5}}
=> {{1,4,5},{2,3}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3,5},{2,4}}
=> {{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1,3},{2,4,5}}
=> {{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1,3},{2,4},{5}}
=> {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3,5},{2},{4}}
=> {{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,5},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
{{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
{{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1
{{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
{{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2,5},{3}}
=> {{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2},{3,5}}
=> {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,4},{3,5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> [6] => ([],6)
=> ? = 6
{{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> [1,5] => ([(4,5)],6)
=> ? = 5
{{1,2,3,4,6},{5}}
=> {{1,3},{2,4,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 4
{{1,2,3,4},{5,6}}
=> {{1,2},{3,4,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 4
{{1,2,3,4},{5},{6}}
=> {{1},{2},{3,4,5,6}}
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 4
{{1,2,3,5,6},{4}}
=> {{1,2,4},{3,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3
{{1,2,3,5},{4,6}}
=> {{1,3,4},{2,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3
{{1,2,3,5},{4},{6}}
=> {{1},{2,4},{3,5,6}}
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
{{1,2,3,6},{4,5}}
=> {{1,4},{2,3,5,6}}
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 3
{{1,2,3},{4,5,6}}
=> {{1,2,3},{4,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3
{{1,2,3},{4,5},{6}}
=> {{1},{2,3},{4,5,6}}
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
{{1,2,3,6},{4},{5}}
=> {{1,4},{2},{3,5,6}}
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
{{1,2,3},{4,6},{5}}
=> {{1,3},{2},{4,5,6}}
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
{{1,2,3},{4},{5,6}}
=> {{1,2},{3},{4,5,6}}
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
{{1,2,3},{4},{5},{6}}
=> {{1},{2},{3},{4,5,6}}
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
{{1,2,4,5,6},{3}}
=> {{1,2,3,5},{4,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
{{1,2,4,5},{3,6}}
=> {{1,4,5},{2,3,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
{{1,2,4,5},{3},{6}}
=> {{1},{2,3,5},{4,6}}
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
{{1,2,4,6},{3,5}}
=> {{1,3,4,5},{2,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
{{1,2,4},{3,5,6}}
=> {{1,2,4,5},{3,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
{{1,3,4,5,6},{2}}
=> {{1,2,3,4,6},{5}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
{{1,3,4,5},{2},{6}}
=> {{1},{2,3,4,6},{5}}
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,3,4,6},{2},{5}}
=> {{1,3},{2,4,6},{5}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,3,4},{2},{5,6}}
=> {{1,2},{3,4,6},{5}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,3,4},{2},{5},{6}}
=> {{1},{2},{3,4,6},{5}}
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,3,5,6},{2,4}}
=> {{1,2,4,5,6},{3}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000989
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 1 = 2 - 1
{{1},{2}}
=> {{1,2}}
=> [2,1] => [2,1] => 0 = 1 - 1
{{1,2,3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 2 = 3 - 1
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0 = 1 - 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 0 = 1 - 1
{{1},{2},{3}}
=> {{1,2,3}}
=> [2,3,1] => [3,2,1] => 0 = 1 - 1
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 0 = 1 - 1
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 0 = 1 - 1
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => 0 = 1 - 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 0 = 1 - 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 0 = 1 - 1
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 0 = 1 - 1
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 2 = 3 - 1
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => 2 = 3 - 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => 2 = 3 - 1
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,3,2,5] => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 1 = 2 - 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => 1 = 2 - 1
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,4,1,5] => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,2,1,5] => 1 = 2 - 1
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,3,2,5] => 1 = 2 - 1
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => 1 = 2 - 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,3,1,5] => 1 = 2 - 1
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,3,2,1,5] => 1 = 2 - 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,5,1,4,2] => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,5,2,4,3] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,5,1,4,3] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,4,3] => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,3,1,4,2] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => 0 = 1 - 1
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,1,3,2] => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 0 = 1 - 1
{{1,2,3,4,5,6},{7}}
=> {{1,2},{3},{4},{5},{6},{7}}
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 6 - 1
{{1,2,3,4,5},{6,7}}
=> {{1,3},{2},{4},{5},{6},{7}}
=> [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 5 - 1
{{1,2,3,4,5},{6},{7}}
=> {{1,2,3},{4},{5},{6},{7}}
=> [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 5 - 1
{{1,2,3,4,6},{5,7}}
=> {{1,3},{2,4},{5},{6},{7}}
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 4 - 1
{{1,2,3,4,6},{5},{7}}
=> {{1,2},{3,4},{5},{6},{7}}
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 4 - 1
{{1,2,3,4},{5,6,7}}
=> {{1,4},{2},{3},{5},{6},{7}}
=> [4,2,3,1,5,6,7] => [2,3,4,1,5,6,7] => ? = 4 - 1
{{1,2,3,4},{5,6},{7}}
=> {{1,2,4},{3},{5},{6},{7}}
=> [2,4,3,1,5,6,7] => [3,4,2,1,5,6,7] => ? = 4 - 1
{{1,2,3,4},{5,7},{6}}
=> {{1,4},{2,3},{5},{6},{7}}
=> [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ? = 4 - 1
{{1,2,3,4},{5},{6,7}}
=> {{1,3,4},{2},{5},{6},{7}}
=> [3,2,4,1,5,6,7] => [2,4,3,1,5,6,7] => ? = 4 - 1
{{1,2,3,4},{5},{6},{7}}
=> {{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 4 - 1
{{1,2,3,5,6},{4,7}}
=> {{1,4},{2,5},{3},{6},{7}}
=> [4,5,3,1,2,6,7] => [3,5,1,4,2,6,7] => ? = 3 - 1
{{1,2,3,5,6},{4},{7}}
=> {{1,2},{3},{4,5},{6},{7}}
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 3 - 1
{{1,2,3,5,7},{4,6}}
=> {{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,5,2,4,3,6,7] => ? = 3 - 1
{{1,2,3,5},{4,6,7}}
=> {{1,4},{2},{3,5},{6},{7}}
=> [4,2,5,1,3,6,7] => [2,5,1,4,3,6,7] => ? = 3 - 1
{{1,2,3,5},{4,6},{7}}
=> {{1,2,4},{3,5},{6},{7}}
=> [2,4,5,1,3,6,7] => [5,2,1,4,3,6,7] => ? = 3 - 1
{{1,2,3,5},{4,7},{6}}
=> {{1,4},{2,3,5},{6},{7}}
=> [4,3,5,1,2,6,7] => [5,3,1,4,2,6,7] => ? = 3 - 1
{{1,2,3,5},{4},{6,7}}
=> {{1,3},{2},{4,5},{6},{7}}
=> [3,2,1,5,4,6,7] => [2,3,1,5,4,6,7] => ? = 3 - 1
{{1,2,3,5},{4},{6},{7}}
=> {{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 3 - 1
{{1,2,3,6},{4,5,7}}
=> {{1,3},{2,5},{4},{6},{7}}
=> [3,5,1,4,2,6,7] => [4,5,1,3,2,6,7] => ? = 3 - 1
{{1,2,3,6},{4,5},{7}}
=> {{1,2},{3,5},{4},{6},{7}}
=> [2,1,5,4,3,6,7] => [2,1,4,5,3,6,7] => ? = 3 - 1
{{1,2,3},{4,5,6,7}}
=> {{1,5},{2},{3},{4},{6},{7}}
=> [5,2,3,4,1,6,7] => [2,3,4,5,1,6,7] => ? = 3 - 1
{{1,2,3},{4,5,6},{7}}
=> {{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [3,4,5,2,1,6,7] => ? = 3 - 1
{{1,2,3},{4,5,7},{6}}
=> {{1,5},{2,3},{4},{6},{7}}
=> [5,3,2,4,1,6,7] => [3,2,4,5,1,6,7] => ? = 3 - 1
{{1,2,3},{4,5},{6,7}}
=> {{1,3,5},{2},{4},{6},{7}}
=> [3,2,5,4,1,6,7] => [2,4,5,3,1,6,7] => ? = 3 - 1
{{1,2,3},{4,5},{6},{7}}
=> {{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [4,5,3,2,1,6,7] => ? = 3 - 1
{{1,2,3,6},{4,7},{5}}
=> {{1,3,4},{2,5},{6},{7}}
=> [3,5,4,1,2,6,7] => [4,1,5,3,2,6,7] => ? = 3 - 1
{{1,2,3,6},{4},{5,7}}
=> {{1,3},{2,4,5},{6},{7}}
=> [3,4,1,5,2,6,7] => [5,4,1,3,2,6,7] => ? = 3 - 1
{{1,2,3,6},{4},{5},{7}}
=> {{1,2},{3,4,5},{6},{7}}
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => ? = 3 - 1
{{1,2,3},{4,6,7},{5}}
=> {{1,5},{2},{3,4},{6},{7}}
=> [5,2,4,3,1,6,7] => [2,4,3,5,1,6,7] => ? = 3 - 1
{{1,2,3},{4,6},{5,7}}
=> {{1,3,5},{2,4},{6},{7}}
=> [3,4,5,2,1,6,7] => [5,2,4,3,1,6,7] => ? = 3 - 1
{{1,2,3},{4,6},{5},{7}}
=> {{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [4,3,5,2,1,6,7] => ? = 3 - 1
{{1,2,3},{4,7},{5,6}}
=> {{1,5},{2,4},{3},{6},{7}}
=> [5,4,3,2,1,6,7] => [3,4,2,5,1,6,7] => ? = 3 - 1
{{1,2,3},{4},{5,6,7}}
=> {{1,4,5},{2},{3},{6},{7}}
=> [4,2,3,5,1,6,7] => [2,3,5,4,1,6,7] => ? = 3 - 1
{{1,2,3},{4},{5,6},{7}}
=> {{1,2,4,5},{3},{6},{7}}
=> [2,4,3,5,1,6,7] => [3,5,4,2,1,6,7] => ? = 3 - 1
{{1,2,3,7},{4},{5},{6}}
=> {{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => ? = 3 - 1
{{1,2,3},{4,7},{5},{6}}
=> {{1,5},{2,3,4},{6},{7}}
=> [5,3,4,2,1,6,7] => [4,3,2,5,1,6,7] => ? = 3 - 1
{{1,2,3},{4},{5,7},{6}}
=> {{1,4,5},{2,3},{6},{7}}
=> [4,3,2,5,1,6,7] => [3,2,5,4,1,6,7] => ? = 3 - 1
{{1,2,3},{4},{5},{6,7}}
=> {{1,3,4,5},{2},{6},{7}}
=> [3,2,4,5,1,6,7] => [2,5,4,3,1,6,7] => ? = 3 - 1
{{1,2,3},{4},{5},{6},{7}}
=> {{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [5,4,3,2,1,6,7] => ? = 3 - 1
{{1,2,4,5,6},{3,7}}
=> {{1,5},{2,6},{3},{4},{7}}
=> [5,6,3,4,1,2,7] => [3,4,6,1,5,2,7] => ? = 2 - 1
{{1,2,4,5,6},{3},{7}}
=> {{1,2},{3},{4},{5,6},{7}}
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2 - 1
{{1,2,4,5},{3,6,7}}
=> {{1,5},{2},{3,6},{4},{7}}
=> [5,2,6,4,1,3,7] => [2,4,6,1,5,3,7] => ? = 2 - 1
{{1,2,4,5},{3,6},{7}}
=> {{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [4,6,2,1,5,3,7] => ? = 2 - 1
{{1,2,4,5},{3,7},{6}}
=> {{1,5},{2,3,6},{4},{7}}
=> [5,3,6,4,1,2,7] => [4,6,3,1,5,2,7] => ? = 2 - 1
{{1,2,4,5},{3},{6,7}}
=> {{1,3},{2},{4},{5,6},{7}}
=> [3,2,1,4,6,5,7] => [2,3,1,4,6,5,7] => ? = 2 - 1
{{1,2,4,5},{3},{6},{7}}
=> {{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 2 - 1
{{1,2,4,6},{3,5,7}}
=> {{1,3,5},{2,4,6},{7}}
=> [3,4,5,6,1,2,7] => [6,1,5,4,3,2,7] => ? = 2 - 1
{{1,2,4,6},{3,5},{7}}
=> {{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [2,1,6,3,5,4,7] => ? = 2 - 1
{{1,2,4},{3,5,6,7}}
=> {{1,5},{2},{3},{4,6},{7}}
=> [5,2,3,6,1,4,7] => [2,3,6,1,5,4,7] => ? = 2 - 1
{{1,2,4},{3,5,6},{7}}
=> {{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [3,6,2,1,5,4,7] => ? = 2 - 1
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St000654
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [[1,2]]
=> [1,2] => [1,2] => 2
{{1},{2}}
=> [[1],[2]]
=> [2,1] => [2,1] => 1
{{1,2,3}}
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 3
{{1,2},{3}}
=> [[1,2],[3]]
=> [3,1,2] => [2,3,1] => 2
{{1,3},{2}}
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 4
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 3
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 2
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 2
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 1
{{1,2,3,4,5}}
=> [[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]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 4
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => 3
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 3
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 3
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => 2
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => 2
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => 2
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => 2
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => 2
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => 2
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => 2
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => 2
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => 2
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => 2
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => 1
{{1,2,3,4,5,6},{7}}
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 6
{{1,2,3,4,5,7},{6}}
=> [[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 5
{{1,2,3,4,5},{6,7}}
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [3,4,5,6,7,1,2] => ? = 5
{{1,2,3,4,5},{6},{7}}
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 5
{{1,2,3,4,6,7},{5}}
=> [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
{{1,2,3,4,6},{5,7}}
=> [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [3,4,5,6,1,7,2] => ? = 4
{{1,2,3,4,6},{5},{7}}
=> [[1,2,3,4,6],[5],[7]]
=> [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 4
{{1,2,3,4,7},{5,6}}
=> [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 4
{{1,2,3,4},{5,6,7}}
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 4
{{1,2,3,4},{5,6},{7}}
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [4,5,6,7,2,3,1] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [3,4,5,6,2,1,7] => ? = 4
{{1,2,3,4},{5,7},{6}}
=> [[1,2,3,4],[5,7],[6]]
=> [6,5,7,1,2,3,4] => [4,5,6,7,2,1,3] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [[1,2,3,4],[5,7],[6]]
=> [6,5,7,1,2,3,4] => [4,5,6,7,2,1,3] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ? = 3
{{1,2,3,5,6},{4,7}}
=> [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [3,4,5,1,6,7,2] => ? = 3
{{1,2,3,5,6},{4},{7}}
=> [[1,2,3,5,6],[4],[7]]
=> [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => ? = 3
{{1,2,3,5,7},{4,6}}
=> [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 3
{{1,2,3,5},{4,6,7}}
=> [[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => ? = 3
{{1,2,3,5},{4,6},{7}}
=> [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => [4,5,6,2,7,3,1] => ? = 3
{{1,2,3,5,7},{4},{6}}
=> [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [3,4,5,2,6,1,7] => ? = 3
{{1,2,3,5},{4,7},{6}}
=> [[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [4,5,6,2,7,1,3] => ? = 3
{{1,2,3,5},{4},{6,7}}
=> [[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [4,5,6,2,7,1,3] => ? = 3
{{1,2,3,5},{4},{6},{7}}
=> [[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [4,5,6,3,7,2,1] => ? = 3
{{1,2,3,6,7},{4,5}}
=> [[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [3,4,5,1,2,6,7] => ? = 3
{{1,2,3,6},{4,5,7}}
=> [[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [4,5,6,1,2,7,3] => ? = 3
{{1,2,3,6},{4,5},{7}}
=> [[1,2,3,6],[4,5],[7]]
=> [7,4,5,1,2,3,6] => [4,5,6,2,3,7,1] => ? = 3
{{1,2,3,7},{4,5,6}}
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 3
{{1,2,3},{4,5,6,7}}
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 3
{{1,2,3},{4,5,6},{7}}
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [5,6,7,2,3,4,1] => ? = 3
{{1,2,3,7},{4,5},{6}}
=> [[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [4,5,6,2,3,1,7] => ? = 3
{{1,2,3},{4,5,7},{6}}
=> [[1,2,3],[4,5,7],[6]]
=> [6,4,5,7,1,2,3] => [5,6,7,2,3,1,4] => ? = 3
{{1,2,3},{4,5},{6,7}}
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [5,6,7,3,4,1,2] => ? = 3
{{1,2,3},{4,5},{6},{7}}
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [5,6,7,3,4,2,1] => ? = 3
{{1,2,3,6,7},{4},{5}}
=> [[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [3,4,5,2,1,6,7] => ? = 3
{{1,2,3,6},{4,7},{5}}
=> [[1,2,3,6],[4,7],[5]]
=> [5,4,7,1,2,3,6] => [4,5,6,2,1,7,3] => ? = 3
{{1,2,3,6},{4},{5,7}}
=> [[1,2,3,6],[4,7],[5]]
=> [5,4,7,1,2,3,6] => [4,5,6,2,1,7,3] => ? = 3
{{1,2,3,6},{4},{5},{7}}
=> [[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [4,5,6,3,2,7,1] => ? = 3
{{1,2,3,7},{4,6},{5}}
=> [[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [4,5,6,2,1,3,7] => ? = 3
{{1,2,3},{4,6,7},{5}}
=> [[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [5,6,7,2,1,3,4] => ? = 3
{{1,2,3},{4,6},{5,7}}
=> [[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [5,6,7,3,1,4,2] => ? = 3
{{1,2,3},{4,6},{5},{7}}
=> [[1,2,3],[4,6],[5],[7]]
=> [7,5,4,6,1,2,3] => [5,6,7,3,2,4,1] => ? = 3
{{1,2,3,7},{4},{5,6}}
=> [[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [4,5,6,2,1,3,7] => ? = 3
{{1,2,3},{4,7},{5,6}}
=> [[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [5,6,7,3,1,4,2] => ? = 3
{{1,2,3},{4},{5,6,7}}
=> [[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [5,6,7,2,1,3,4] => ? = 3
{{1,2,3},{4},{5,6},{7}}
=> [[1,2,3],[4,6],[5],[7]]
=> [7,5,4,6,1,2,3] => [5,6,7,3,2,4,1] => ? = 3
{{1,2,3,7},{4},{5},{6}}
=> [[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [4,5,6,3,2,1,7] => ? = 3
{{1,2,3},{4,7},{5},{6}}
=> [[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [5,6,7,3,2,1,4] => ? = 3
{{1,2,3},{4},{5,7},{6}}
=> [[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [5,6,7,3,2,1,4] => ? = 3
{{1,2,3},{4},{5},{6,7}}
=> [[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [5,6,7,3,2,1,4] => ? = 3
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
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