searching the database
Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000745
(load all 64 compositions to match this statistic)
(load all 64 compositions to match this statistic)
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [[1,2]]
=> 1 = 0 + 1
{{1},{2}}
=> [[1],[2]]
=> 2 = 1 + 1
{{1,2,3}}
=> [[1,2,3]]
=> 1 = 0 + 1
{{1,2},{3}}
=> [[1,2],[3]]
=> 1 = 0 + 1
{{1,3},{2}}
=> [[1,3],[2]]
=> 2 = 1 + 1
{{1},{2,3}}
=> [[1,3],[2]]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> 3 = 2 + 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> 2 = 1 + 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> 2 = 1 + 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> 2 = 1 + 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> 2 = 1 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000297
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ 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%
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]]
=> 0 => 0
{{1},{2}}
=> [[1],[2]]
=> 1 => 1
{{1,2,3}}
=> [[1,2,3]]
=> 00 => 0
{{1,2},{3}}
=> [[1,2],[3]]
=> 01 => 0
{{1,3},{2}}
=> [[1,3],[2]]
=> 10 => 1
{{1},{2,3}}
=> [[1,3],[2]]
=> 10 => 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> 11 => 2
{{1,2,3,4}}
=> [[1,2,3,4]]
=> 000 => 0
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> 001 => 0
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> 010 => 0
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> 010 => 0
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> 011 => 0
{{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 => 2
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> 110 => 2
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> 110 => 2
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> 111 => 3
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> 0000 => 0
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> 0001 => 0
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> 0010 => 0
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> 0010 => 0
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> 0011 => 0
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> 0100 => 0
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> 0101 => 0
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> 0101 => 0
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> 0100 => 0
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> 0100 => 0
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> 0101 => 0
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> 0110 => 0
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> 0110 => 0
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> 0110 => 0
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> 0111 => 0
{{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 number of leading ones in a binary word.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 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
St000326: 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
St000326: 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 = 0 + 1
{{1},{2}}
=> [[1],[2]]
=> [[1,2]]
=> 0 => 2 = 1 + 1
{{1,2,3}}
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 11 => 1 = 0 + 1
{{1,2},{3}}
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 10 => 1 = 0 + 1
{{1,3},{2}}
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 01 => 2 = 1 + 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 01 => 2 = 1 + 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 00 => 3 = 2 + 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 111 => 1 = 0 + 1
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 110 => 1 = 0 + 1
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 101 => 1 = 0 + 1
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 101 => 1 = 0 + 1
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 100 => 1 = 0 + 1
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 011 => 2 = 1 + 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 010 => 2 = 1 + 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 010 => 2 = 1 + 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 010 => 2 = 1 + 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 011 => 2 = 1 + 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 010 => 2 = 1 + 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 3 = 2 + 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 3 = 2 + 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 3 = 2 + 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 000 => 4 = 3 + 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1110 => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 1101 => 1 = 0 + 1
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 1101 => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1100 => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 1011 => 1 = 0 + 1
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 1010 => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 1010 => 1 = 0 + 1
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 1011 => 1 = 0 + 1
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 1011 => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1010 => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 1001 => 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 1001 => 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 1001 => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1000 => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0111 => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 0110 => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 0110 => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 0100 => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 0101 => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 0101 => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 0101 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 0100 => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 0100 => 2 = 1 + 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 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ 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%
Mp00084: Standard tableaux —conjugate⟶ 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%
Values
{{1,2}}
=> [[1,2]]
=> [[1],[2]]
=> [1,1] => 1 = 0 + 1
{{1},{2}}
=> [[1],[2]]
=> [[1,2]]
=> [2] => 2 = 1 + 1
{{1,2,3}}
=> [[1,2,3]]
=> [[1],[2],[3]]
=> [1,1,1] => 1 = 0 + 1
{{1,2},{3}}
=> [[1,2],[3]]
=> [[1,3],[2]]
=> [1,2] => 1 = 0 + 1
{{1,3},{2}}
=> [[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => 2 = 1 + 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => 2 = 1 + 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [3] => 3 = 2 + 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1 = 0 + 1
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> [1,2,1] => 1 = 0 + 1
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [1,2,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [1,3] => 1 = 0 + 1
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2 = 1 + 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> [2,2] => 2 = 1 + 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> [2,2] => 2 = 1 + 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2 = 1 + 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> [3,1] => 3 = 2 + 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> [3,1] => 3 = 2 + 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> [3,1] => 3 = 2 + 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [4] => 4 = 3 + 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> [1,1,2,1] => 1 = 0 + 1
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [1,1,2,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => 1 = 0 + 1
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> [1,2,2] => 1 = 0 + 1
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => 1 = 0 + 1
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [1,2,2] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> [1,3,1] => 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> [1,3,1] => 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> [1,3,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [1,4] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> [2,1,2] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> [2,3] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> [2,3] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> [2,3] => 2 = 1 + 1
Description
The first part of an integer composition.
Matching statistic: St000383
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [[1,2]]
=> [2] => [1,1] => 1 = 0 + 1
{{1},{2}}
=> [[1],[2]]
=> [1,1] => [2] => 2 = 1 + 1
{{1,2,3}}
=> [[1,2,3]]
=> [3] => [1,1,1] => 1 = 0 + 1
{{1,2},{3}}
=> [[1,2],[3]]
=> [2,1] => [2,1] => 1 = 0 + 1
{{1,3},{2}}
=> [[1,3],[2]]
=> [1,2] => [1,2] => 2 = 1 + 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [1,2] => [1,2] => 2 = 1 + 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3 = 2 + 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [4] => [1,1,1,1] => 1 = 0 + 1
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [3,1] => [2,1,1] => 1 = 0 + 1
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [2,2] => [1,2,1] => 1 = 0 + 1
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [2,2] => [1,2,1] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [2,1,1] => [3,1] => 1 = 0 + 1
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [1,3] => [1,1,2] => 2 = 1 + 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [1,2,1] => [2,2] => 2 = 1 + 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [1,2,1] => [2,2] => 2 = 1 + 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [1,2,1] => [2,2] => 2 = 1 + 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [1,3] => [1,1,2] => 2 = 1 + 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [1,2,1] => [2,2] => 2 = 1 + 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [1,1,2] => [1,3] => 3 = 2 + 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [1,1,2] => [1,3] => 3 = 2 + 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [1,1,2] => [1,3] => 3 = 2 + 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4 = 3 + 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [4,1] => [2,1,1,1] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [3,2] => [1,2,1,1] => 1 = 0 + 1
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [3,2] => [1,2,1,1] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [3,1,1] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [2,3] => [1,1,2,1] => 1 = 0 + 1
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [2,2,1] => [2,2,1] => 1 = 0 + 1
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [2,3] => [1,1,2,1] => 1 = 0 + 1
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [2,3] => [1,1,2,1] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [2,1,2] => [1,3,1] => 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [2,1,2] => [1,3,1] => 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [2,1,2] => [1,3,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [4,1] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [1,4] => [1,1,1,2] => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [1,3,1] => [2,1,2] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [1,3,1] => [2,1,2] => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => [3,2] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [1,2,2] => [1,2,2] => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [1,2,2] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [1,2,2] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => [3,2] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => [3,2] => 2 = 1 + 1
{{1,8},{2,7},{3,5},{4,6}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,7},{2,8},{3,5},{4,6}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,6},{2,8},{3,5},{4,7}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,5},{2,8},{3,6},{4,7}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,5},{2,7},{3,6},{4,8}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,6},{2,7},{3,5},{4,8}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,7},{2,6},{3,5},{4,8}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,8},{2,6},{3,5},{4,7}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,8},{2,5},{3,6},{4,7}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,7},{2,5},{3,6},{4,8}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,6},{2,5},{3,7},{4,8}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,5},{2,6},{3,7},{4,8}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,5},{2,6},{3,8},{4,7}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,6},{2,5},{3,8},{4,7}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,7},{2,5},{3,8},{4,6}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,8},{2,5},{3,7},{4,6}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,8},{2,6},{3,7},{4,5}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,7},{2,6},{3,8},{4,5}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,6},{2,7},{3,8},{4,5}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,5},{2,7},{3,8},{4,6}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,5},{2,8},{3,7},{4,6}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,6},{2,8},{3,7},{4,5}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,7},{2,8},{3,6},{4,5}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,8},{2,7},{3,6},{4,5}}
=> [[1,5],[2,6],[3,7],[4,8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1},{2,6,8},{3,5},{4},{7}}
=> [[1,5,8],[2,6],[3],[4],[7]]
=> [1,1,1,2,1,2] => [1,3,4] => ? = 3 + 1
{{1},{2,8},{3,5,7},{4},{6}}
=> [[1,5,7],[2,8],[3],[4],[6]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1},{2,8},{3,7},{4,6},{5}}
=> [[1,6],[2,7],[3,8],[4],[5]]
=> [1,1,1,1,2,2] => [1,2,5] => ? = 4 + 1
{{1,2},{3,6,7,8},{4},{5}}
=> [[1,2,7,8],[3,6],[4],[5]]
=> [2,1,1,4] => [1,1,1,4,1] => ? = 0 + 1
{{1,5,6,7,8},{2},{3},{4}}
=> [[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => [1,1,1,1,4] => ? = 3 + 1
{{1,6,7,8},{2},{3},{4,5}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,4,5,6},{2},{3},{7,8}}
=> [[1,4,5,6],[2,8],[3],[7]]
=> [1,1,4,2] => [1,2,1,1,3] => ? = 2 + 1
{{1,4,7,8},{2},{3},{5,6}}
=> [[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => [1,1,1,2,3] => ? = 2 + 1
{{1,4,5,8},{2},{3},{6,7}}
=> [[1,4,5,8],[2,7],[3],[6]]
=> [1,1,3,3] => [1,1,2,1,3] => ? = 2 + 1
{{1,6,7,8},{2},{3,5},{4}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,7,8},{2},{3,6},{4,5}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,4},{2,3},{5,7,8},{6}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [1,1,4,2] => ? = 1 + 1
{{1,4,8},{2,3},{5},{6,7}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [1,1,4,2] => ? = 1 + 1
{{1,4,8},{2,3},{5,7},{6}}
=> [[1,3,8],[2,4],[5,7],[6]]
=> [1,2,1,1,3] => [1,1,4,2] => ? = 1 + 1
{{1,6,7,8},{2,5},{3},{4}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,8},{2,5,6,7},{3},{4}}
=> [[1,5,6,7],[2,8],[3],[4]]
=> [1,1,1,4,1] => [2,1,1,4] => ? = 3 + 1
{{1,7,8},{2,6},{3},{4,5}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,8},{2,6,7},{3},{4,5}}
=> [[1,5,7],[2,6],[3,8],[4]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,7},{2,6},{3,5},{4},{8}}
=> [[1,5],[2,6],[3,7],[4],[8]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,7,8},{2,6},{3,5},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,8},{2,6,7},{3,5},{4}}
=> [[1,5,7],[2,6],[3,8],[4]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
{{1,5,7,8},{2,6},{3},{4}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,7,8},{2,5},{3,6},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,5,7,8},{2},{3,6},{4}}
=> [[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,5,8},{2,7},{3,6},{4}}
=> [[1,5,8],[2,6],[3,7],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 3 + 1
{{1,8},{2,5,7},{3,6},{4}}
=> [[1,5,7],[2,6],[3,8],[4]]
=> [1,1,1,2,2,1] => [2,2,4] => ? = 3 + 1
Description
The last part of an integer composition.
Matching statistic: St001050
(load all 217 compositions to match this statistic)
(load all 217 compositions to match this statistic)
Mp00112: Set partitions —complement⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
St001050: Set partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> 1 = 0 + 1
{{1},{2}}
=> {{1},{2}}
=> 2 = 1 + 1
{{1,2,3}}
=> {{1,2,3}}
=> 1 = 0 + 1
{{1,2},{3}}
=> {{1},{2,3}}
=> 1 = 0 + 1
{{1,3},{2}}
=> {{1,3},{2}}
=> 2 = 1 + 1
{{1},{2,3}}
=> {{1,2},{3}}
=> 2 = 1 + 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 3 = 2 + 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 1 = 0 + 1
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1 = 0 + 1
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1 = 0 + 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 2 = 1 + 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 2 = 1 + 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2 = 1 + 1
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3 = 2 + 1
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2 = 1 + 1
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 2 = 1 + 1
{{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 2 = 1 + 1
{{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 2 = 1 + 1
{{1},{2,3},{4,5,6,8},{7}}
=> {{1,3,4,5},{2},{6,7},{8}}
=> ? = 1 + 1
{{1},{2,4,6,8},{3},{5},{7}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 2 + 1
{{1},{2,4,8},{3},{5,7},{6}}
=> {{1,5,7},{2,4},{3},{6},{8}}
=> ? = 2 + 1
{{1},{2,8},{3,4},{5,6,7}}
=> {{1,7},{2,3,4},{5,6},{8}}
=> ? = 2 + 1
{{1},{2,3,5},{4},{6,7,8}}
=> {{1,2,3},{4,6,7},{5},{8}}
=> ? = 1 + 1
{{1},{2,6,8},{3,5},{4},{7}}
=> {{1,3,7},{2},{4,6},{5},{8}}
=> ? = 3 + 1
{{1},{2,8},{3,5,7},{4},{6}}
=> {{1,7},{2,4,6},{3},{5},{8}}
=> ? = 3 + 1
{{1},{2,8},{3,7},{4,6},{5}}
=> {{1,7},{2,6},{3,5},{4},{8}}
=> ? = 4 + 1
{{1,2},{3,4},{5,7,8},{6}}
=> {{1,2,4},{3},{5,6},{7,8}}
=> ? = 0 + 1
{{1,2},{3,4},{5,6,8},{7}}
=> {{1,3,4},{2},{5,6},{7,8}}
=> ? = 0 + 1
{{1,2},{3,6,7,8},{4},{5}}
=> {{1,2,3,6},{4},{5},{7,8}}
=> ? = 0 + 1
{{1,2},{3,5,6},{4},{7,8}}
=> {{1,2},{3,4,6},{5},{7,8}}
=> ? = 0 + 1
{{1,2},{3,7,8},{4},{5,6}}
=> {{1,2,6},{3,4},{5},{7,8}}
=> ? = 0 + 1
{{1,2},{3,5,8},{4},{6,7}}
=> {{1,4,6},{2,3},{5},{7,8}}
=> ? = 0 + 1
{{1,2},{3,5,6,8},{4},{7}}
=> {{1,3,4,6},{2},{5},{7,8}}
=> ? = 0 + 1
{{1,2},{3,6,8},{4,5},{7}}
=> {{1,3,6},{2},{4,5},{7,8}}
=> ? = 0 + 1
{{1,2},{3,4,6},{5},{7,8}}
=> {{1,2},{3,5,6},{4},{7,8}}
=> ? = 0 + 1
{{1,2},{3,7,8},{4,6},{5}}
=> {{1,2,6},{3,5},{4},{7,8}}
=> ? = 0 + 1
{{1,2},{3,8},{4,6,7},{5}}
=> {{1,6},{2,3,5},{4},{7,8}}
=> ? = 0 + 1
{{1,2},{3,4,8},{5},{6,7}}
=> {{1,5,6},{2,3},{4},{7,8}}
=> ? = 0 + 1
{{1,2},{3,4,6,8},{5},{7}}
=> {{1,3,5,6},{2},{4},{7,8}}
=> ? = 0 + 1
{{1,2},{3,8},{4,5,7},{6}}
=> {{1,6},{2,4,5},{3},{7,8}}
=> ? = 0 + 1
{{1,2},{3,4,8},{5,7},{6}}
=> {{1,5,6},{2,4},{3},{7,8}}
=> ? = 0 + 1
{{1,3},{2},{4,6,8},{5},{7}}
=> {{1,3,5},{2},{4},{6,8},{7}}
=> ? = 1 + 1
{{1,3},{2},{4,8},{5,7},{6}}
=> {{1,5},{2,4},{3},{6,8},{7}}
=> ? = 1 + 1
{{1,3},{2},{4,5,6,8},{7}}
=> {{1,3,4,5},{2},{6,8},{7}}
=> ? = 1 + 1
{{1,5,6,7,8},{2},{3},{4}}
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 3 + 1
{{1,6,7,8},{2},{3},{4,5}}
=> {{1,2,3,8},{4,5},{6},{7}}
=> ? = 3 + 1
{{1,4,5,6},{2},{3},{7,8}}
=> {{1,2},{3,4,5,8},{6},{7}}
=> ? = 2 + 1
{{1,4,7,8},{2},{3},{5,6}}
=> {{1,2,5,8},{3,4},{6},{7}}
=> ? = 2 + 1
{{1,4,5,8},{2},{3},{6,7}}
=> {{1,4,5,8},{2,3},{6},{7}}
=> ? = 2 + 1
{{1,3,4},{2},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6,8},{7}}
=> ? = 1 + 1
{{1,3,4},{2},{5,7,8},{6}}
=> {{1,2,4},{3},{5,6,8},{7}}
=> ? = 1 + 1
{{1,3,4},{2},{5,8},{6,7}}
=> {{1,4},{2,3},{5,6,8},{7}}
=> ? = 1 + 1
{{1,5,6},{2},{3,4},{7,8}}
=> {{1,2},{3,4,8},{5,6},{7}}
=> ? = 2 + 1
{{1,7,8},{2},{3,4},{5,6}}
=> {{1,2,8},{3,4},{5,6},{7}}
=> ? = 2 + 1
{{1,5,8},{2},{3,4},{6,7}}
=> {{1,4,8},{2,3},{5,6},{7}}
=> ? = 2 + 1
{{1,3,5},{2},{4},{6,8},{7}}
=> {{1,3},{2},{4,6,8},{5},{7}}
=> ? = 1 + 1
{{1,3,5},{2},{4},{6,7,8}}
=> {{1,2,3},{4,6,8},{5},{7}}
=> ? = 1 + 1
{{1,6,7,8},{2},{3,5},{4}}
=> {{1,2,3,8},{4,6},{5},{7}}
=> ? = 3 + 1
{{1,7,8},{2},{3,5,6},{4}}
=> {{1,2,8},{3,4,6},{5},{7}}
=> ? = 3 + 1
{{1,3,5,7},{2},{4},{6},{8}}
=> {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 1 + 1
{{1,3,6},{2},{4,5},{7,8}}
=> {{1,2},{3,6,8},{4,5},{7}}
=> ? = 1 + 1
{{1,7,8},{2},{3,6},{4,5}}
=> {{1,2,8},{3,6},{4,5},{7}}
=> ? = 3 + 1
{{1,3,8},{2},{4,5},{6,7}}
=> {{1,6,8},{2,3},{4,5},{7}}
=> ? = 1 + 1
{{1,3,7},{2},{4,6},{5},{8}}
=> {{1},{2,6,8},{3,5},{4},{7}}
=> ? = 1 + 1
{{1,3,8},{2},{4,6,7},{5}}
=> {{1,6,8},{2,3,5},{4},{7}}
=> ? = 1 + 1
{{1,3,8},{2},{4,7},{5,6}}
=> {{1,6,8},{2,5},{3,4},{7}}
=> ? = 1 + 1
{{1,4},{2,3},{5,7,8},{6}}
=> {{1,2,4},{3},{5,8},{6,7}}
=> ? = 1 + 1
{{1,4},{2,3},{5,6,8},{7}}
=> {{1,3,4},{2},{5,8},{6,7}}
=> ? = 1 + 1
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000971
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => {{1},{2}}
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [2,1] => {{1,2}}
=> 2 = 1 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => {{1},{2,3}}
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => {{1,2,3}}
=> 3 = 2 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => {{1},{2,3,4}}
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => {{1,4},{2},{3}}
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => {{1,2,3},{4}}
=> 3 = 2 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> 3 = 2 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 4 = 3 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => {{1,5},{2},{3,4}}
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2 = 1 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 2 = 1 + 1
{{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => [1,4,3,6,5,8,7,2] => {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 0 + 1
{{1,3},{2,4},{5,6},{7,8}}
=> [3,4,1,2,6,5,8,7] => [4,1,2,6,5,8,7,3] => {{1,4,6,8},{2},{3},{5},{7}}
=> ? = 1 + 1
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [3,2,1,6,5,8,7,4] => {{1,3},{2},{4,6,8},{5},{7}}
=> ? = 1 + 1
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => [3,2,6,1,4,8,7,5] => {{1,3,6,8},{2},{4},{5},{7}}
=> ? = 1 + 1
{{1,6},{2,3},{4,5},{7,8}}
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,1,8,7,6] => {{1,3,5},{2},{4},{6,8},{7}}
=> ? = 1 + 1
{{1,7},{2,3},{4,5},{6,8}}
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,8,1,6,7] => {{1,3,5,8},{2},{4},{6},{7}}
=> ? = 1 + 1
{{1,8},{2,3},{4,5},{6,7}}
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,1,8] => {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 1 + 1
{{1,8},{2,4},{3,5},{6,7}}
=> [8,4,5,2,3,7,6,1] => [4,5,2,3,7,6,1,8] => {{1,4},{2,5,7},{3},{6},{8}}
=> ? = 2 + 1
{{1,7},{2,4},{3,5},{6,8}}
=> [7,4,5,2,3,8,1,6] => [4,5,2,3,8,1,6,7] => {{1,4},{2,5,8},{3},{6},{7}}
=> ? = 2 + 1
{{1,6},{2,4},{3,5},{7,8}}
=> [6,4,5,2,3,1,8,7] => [4,5,2,3,1,8,7,6] => {{1,4},{2,5},{3},{6,8},{7}}
=> ? = 2 + 1
{{1,5},{2,4},{3,6},{7,8}}
=> [5,4,6,2,1,3,8,7] => [4,6,2,1,3,8,7,5] => {{1,4},{2,6,8},{3},{5},{7}}
=> ? = 2 + 1
{{1,4},{2,5},{3,6},{7,8}}
=> [4,5,6,1,2,3,8,7] => [5,6,1,2,3,8,7,4] => {{1,5},{2,6,8},{3},{4},{7}}
=> ? = 2 + 1
{{1,3},{2,5},{4,6},{7,8}}
=> [3,5,1,6,2,4,8,7] => [5,1,6,2,4,8,7,3] => {{1,5},{2},{3,6,8},{4},{7}}
=> ? = 1 + 1
{{1,2},{3,5},{4,6},{7,8}}
=> [2,1,5,6,3,4,8,7] => [1,5,6,3,4,8,7,2] => {{1},{2,5},{3,6,8},{4},{7}}
=> ? = 0 + 1
{{1,2},{3,6},{4,5},{7,8}}
=> [2,1,6,5,4,3,8,7] => [1,6,5,4,3,8,7,2] => {{1},{2,6,8},{3,5},{4},{7}}
=> ? = 0 + 1
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => [6,1,5,4,2,8,7,3] => {{1,6,8},{2},{3,5},{4},{7}}
=> ? = 1 + 1
{{1,4},{2,6},{3,5},{7,8}}
=> [4,6,5,1,3,2,8,7] => [6,5,1,3,2,8,7,4] => {{1,6,8},{2,5},{3},{4},{7}}
=> ? = 2 + 1
{{1,5},{2,6},{3,4},{7,8}}
=> [5,6,4,3,1,2,8,7] => [6,4,3,1,2,8,7,5] => {{1,6,8},{2,4},{3},{5},{7}}
=> ? = 2 + 1
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => [5,4,3,2,1,8,7,6] => {{1,5},{2,4},{3},{6,8},{7}}
=> ? = 2 + 1
{{1,7},{2,5},{3,4},{6,8}}
=> [7,5,4,3,2,8,1,6] => [5,4,3,2,8,1,6,7] => {{1,5,8},{2,4},{3},{6},{7}}
=> ? = 2 + 1
{{1,8},{2,5},{3,4},{6,7}}
=> [8,5,4,3,2,7,6,1] => [5,4,3,2,7,6,1,8] => {{1,5,7},{2,4},{3},{6},{8}}
=> ? = 2 + 1
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => [6,4,3,7,2,5,1,8] => {{1,6},{2,4,7},{3},{5},{8}}
=> ? = 2 + 1
{{1,7},{2,6},{3,4},{5,8}}
=> [7,6,4,3,8,2,1,5] => [6,4,3,8,2,1,5,7] => {{1,6},{2,4,8},{3},{5},{7}}
=> ? = 2 + 1
{{1,6},{2,7},{3,4},{5,8}}
=> [6,7,4,3,8,1,2,5] => [7,4,3,8,1,2,5,6] => {{1,7},{2,4,8},{3},{5},{6}}
=> ? = 2 + 1
{{1,5},{2,7},{3,4},{6,8}}
=> [5,7,4,3,1,8,2,6] => [7,4,3,1,8,2,6,5] => {{1,7},{2,4},{3},{5,8},{6}}
=> ? = 2 + 1
{{1,4},{2,7},{3,5},{6,8}}
=> [4,7,5,1,3,8,2,6] => [7,5,1,3,8,2,6,4] => {{1,7},{2,5,8},{3},{4},{6}}
=> ? = 2 + 1
{{1,3},{2,7},{4,5},{6,8}}
=> [3,7,1,5,4,8,2,6] => [7,1,5,4,8,2,6,3] => {{1,7},{2},{3,5,8},{4},{6}}
=> ? = 1 + 1
{{1,2},{3,7},{4,5},{6,8}}
=> [2,1,7,5,4,8,3,6] => [1,7,5,4,8,3,6,2] => {{1},{2,7},{3,5,8},{4},{6}}
=> ? = 0 + 1
{{1,2},{3,8},{4,5},{6,7}}
=> [2,1,8,5,4,7,6,3] => [1,8,5,4,7,6,3,2] => {{1},{2,8},{3,5,7},{4},{6}}
=> ? = 0 + 1
{{1,3},{2,8},{4,5},{6,7}}
=> [3,8,1,5,4,7,6,2] => [8,1,5,4,7,6,2,3] => {{1,8},{2},{3,5,7},{4},{6}}
=> ? = 1 + 1
{{1,4},{2,8},{3,5},{6,7}}
=> [4,8,5,1,3,7,6,2] => [8,5,1,3,7,6,2,4] => {{1,8},{2,5,7},{3},{4},{6}}
=> ? = 2 + 1
{{1,5},{2,8},{3,4},{6,7}}
=> [5,8,4,3,1,7,6,2] => [8,4,3,1,7,6,2,5] => {{1,8},{2,4},{3},{5,7},{6}}
=> ? = 2 + 1
{{1,6},{2,8},{3,4},{5,7}}
=> [6,8,4,3,7,1,5,2] => [8,4,3,7,1,5,2,6] => {{1,8},{2,4,7},{3},{5},{6}}
=> ? = 2 + 1
{{1,7},{2,8},{3,4},{5,6}}
=> [7,8,4,3,6,5,1,2] => [8,4,3,6,5,1,2,7] => {{1,8},{2,4,6},{3},{5},{7}}
=> ? = 2 + 1
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => [7,4,3,6,5,2,1,8] => {{1,7},{2,4,6},{3},{5},{8}}
=> ? = 2 + 1
{{1,8},{2,7},{3,5},{4,6}}
=> [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1,8] => {{1,7},{2,5},{3,6},{4},{8}}
=> ? = 3 + 1
{{1,7},{2,8},{3,5},{4,6}}
=> [7,8,5,6,3,4,1,2] => [8,5,6,3,4,1,2,7] => {{1,8},{2,5},{3,6},{4},{7}}
=> ? = 3 + 1
{{1,6},{2,8},{3,5},{4,7}}
=> [6,8,5,7,3,1,4,2] => [8,5,7,3,1,4,2,6] => {{1,8},{2,5},{3,7},{4},{6}}
=> ? = 3 + 1
{{1,5},{2,8},{3,6},{4,7}}
=> [5,8,6,7,1,3,4,2] => [8,6,7,1,3,4,2,5] => {{1,8},{2,6},{3,7},{4},{5}}
=> ? = 3 + 1
{{1,4},{2,8},{3,6},{5,7}}
=> [4,8,6,1,7,3,5,2] => [8,6,1,7,3,5,2,4] => {{1,8},{2,6},{3},{4,7},{5}}
=> ? = 2 + 1
{{1,3},{2,8},{4,6},{5,7}}
=> [3,8,1,6,7,4,5,2] => [8,1,6,7,4,5,2,3] => {{1,8},{2},{3,6},{4,7},{5}}
=> ? = 1 + 1
{{1,2},{3,8},{4,6},{5,7}}
=> [2,1,8,6,7,4,5,3] => [1,8,6,7,4,5,3,2] => {{1},{2,8},{3,6},{4,7},{5}}
=> ? = 0 + 1
{{1,2},{3,7},{4,6},{5,8}}
=> [2,1,7,6,8,4,3,5] => [1,7,6,8,4,3,5,2] => {{1},{2,7},{3,6},{4,8},{5}}
=> ? = 0 + 1
{{1,3},{2,7},{4,6},{5,8}}
=> [3,7,1,6,8,4,2,5] => [7,1,6,8,4,2,5,3] => {{1,7},{2},{3,6},{4,8},{5}}
=> ? = 1 + 1
{{1,4},{2,7},{3,6},{5,8}}
=> [4,7,6,1,8,3,2,5] => [7,6,1,8,3,2,5,4] => {{1,7},{2,6},{3},{4,8},{5}}
=> ? = 2 + 1
{{1,5},{2,7},{3,6},{4,8}}
=> [5,7,6,8,1,3,2,4] => [7,6,8,1,3,2,4,5] => {{1,7},{2,6},{3,8},{4},{5}}
=> ? = 3 + 1
{{1,6},{2,7},{3,5},{4,8}}
=> [6,7,5,8,3,1,2,4] => [7,5,8,3,1,2,4,6] => {{1,7},{2,5},{3,8},{4},{6}}
=> ? = 3 + 1
{{1,7},{2,6},{3,5},{4,8}}
=> [7,6,5,8,3,2,1,4] => [6,5,8,3,2,1,4,7] => {{1,6},{2,5},{3,8},{4},{7}}
=> ? = 3 + 1
{{1,8},{2,6},{3,5},{4,7}}
=> [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1,8] => {{1,6},{2,5},{3,7},{4},{8}}
=> ? = 3 + 1
{{1,8},{2,5},{3,6},{4,7}}
=> [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1,8] => {{1,5},{2,6},{3,7},{4},{8}}
=> ? = 3 + 1
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St000541
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 86%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
=> [[1,2]]
=> [1,2] => [1,2] => 0
{{1},{2}}
=> [[1],[2]]
=> [2,1] => [2,1] => 1
{{1,2,3}}
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 0
{{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] => 2
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 0
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 0
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 0
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 0
{{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] => [2,1,4,3] => 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => 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] => [2,4,1,3] => 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 2
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 3
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => 0
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => 0
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 0
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 0
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => 0
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => 0
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 0
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 0
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => 0
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => 0
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => 0
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => 0
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 0
{{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] => [2,1,3,5,4] => 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => 1
{{1,2,5},{3},{4},{6},{7}}
=> [[1,2,5],[3],[4],[6],[7]]
=> [7,6,4,3,1,2,5] => [1,4,7,6,3,2,5] => ? = 0
{{1,2},{3,5},{4},{6},{7}}
=> [[1,2],[3,5],[4],[6],[7]]
=> [7,6,4,3,5,1,2] => [1,4,7,6,3,5,2] => ? = 0
{{1,2},{3},{4,5},{6},{7}}
=> [[1,2],[3,5],[4],[6],[7]]
=> [7,6,4,3,5,1,2] => [1,4,7,6,3,5,2] => ? = 0
{{1,2,6,7},{3},{4},{5}}
=> [[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,5,4,3,2,6,7] => ? = 0
{{1,2,6},{3,7},{4},{5}}
=> [[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,5,4,3,2,7,6] => ? = 0
{{1,2,6},{3},{4,7},{5}}
=> [[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,5,4,3,2,7,6] => ? = 0
{{1,2,6},{3},{4},{5,7}}
=> [[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,5,4,3,2,7,6] => ? = 0
{{1,2,6},{3},{4},{5},{7}}
=> [[1,2,6],[3],[4],[5],[7]]
=> [7,5,4,3,1,2,6] => [1,5,7,4,3,2,6] => ? = 0
{{1,2,7},{3,6},{4},{5}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0
{{1,2},{3,6,7},{4},{5}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0
{{1,2},{3,6},{4,7},{5}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0
{{1,2},{3,6},{4},{5,7}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0
{{1,2},{3,6},{4},{5},{7}}
=> [[1,2],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2] => [1,5,7,4,3,6,2] => ? = 0
{{1,2,7},{3},{4,6},{5}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0
{{1,2},{3,7},{4,6},{5}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0
{{1,2},{3},{4,6,7},{5}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0
{{1,2},{3},{4,6},{5,7}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0
{{1,2},{3},{4,6},{5},{7}}
=> [[1,2],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2] => [1,5,7,4,3,6,2] => ? = 0
{{1,2,7},{3},{4},{5,6}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0
{{1,2},{3,7},{4},{5,6}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0
{{1,2},{3},{4,7},{5,6}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0
{{1,2},{3},{4},{5,6,7}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0
{{1,2},{3},{4},{5,6},{7}}
=> [[1,2],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2] => [1,5,7,4,3,6,2] => ? = 0
{{1,2,7},{3},{4},{5},{6}}
=> [[1,2,7],[3],[4],[5],[6]]
=> [6,5,4,3,1,2,7] => [1,6,5,4,3,2,7] => ? = 0
{{1,2},{3,7},{4},{5},{6}}
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [1,6,5,4,3,7,2] => ? = 0
{{1,2},{3},{4,7},{5},{6}}
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [1,6,5,4,3,7,2] => ? = 0
{{1,2},{3},{4},{5,7},{6}}
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [1,6,5,4,3,7,2] => ? = 0
{{1,2},{3},{4},{5},{6,7}}
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [1,6,5,4,3,7,2] => ? = 0
{{1,2},{3},{4},{5},{6},{7}}
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => ? = 0
{{1,3,4,5,6,7},{2}}
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
{{1,3,4,5,6},{2,7}}
=> [[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [2,1,3,4,5,7,6] => ? = 1
{{1,3,4,5,6},{2},{7}}
=> [[1,3,4,5,6],[2],[7]]
=> [7,2,1,3,4,5,6] => [2,1,3,4,7,5,6] => ? = 1
{{1,3,4,5,7},{2,6}}
=> [[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [2,1,3,4,6,5,7] => ? = 1
{{1,3,4,5},{2,6,7}}
=> [[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [2,1,3,4,6,7,5] => ? = 1
{{1,3,4,5},{2,6},{7}}
=> [[1,3,4,5],[2,6],[7]]
=> [7,2,6,1,3,4,5] => [2,1,3,4,7,6,5] => ? = 1
{{1,3,4,5,7},{2},{6}}
=> [[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [2,1,3,6,4,5,7] => ? = 1
{{1,3,4,5},{2,7},{6}}
=> [[1,3,4,5],[2,7],[6]]
=> [6,2,7,1,3,4,5] => [2,1,3,6,4,7,5] => ? = 1
{{1,3,4,5},{2},{6,7}}
=> [[1,3,4,5],[2,7],[6]]
=> [6,2,7,1,3,4,5] => [2,1,3,6,4,7,5] => ? = 1
{{1,3,4,5},{2},{6},{7}}
=> [[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [2,1,3,7,6,4,5] => ? = 1
{{1,3,4,6,7},{2,5}}
=> [[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => ? = 1
{{1,3,4,6},{2,5,7}}
=> [[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => ? = 1
{{1,3,4,6},{2,5},{7}}
=> [[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => ? = 1
{{1,3,4,7},{2,5,6}}
=> [[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [2,1,3,5,6,4,7] => ? = 1
{{1,3,4},{2,5,6,7}}
=> [[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [2,1,3,5,6,4,7] => ? = 1
{{1,3,4},{2,5,6},{7}}
=> [[1,3,4],[2,5,6],[7]]
=> [7,2,5,6,1,3,4] => [2,1,3,5,7,6,4] => ? = 1
{{1,3,4,7},{2,5},{6}}
=> [[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [2,1,3,6,5,4,7] => ? = 1
{{1,3,4},{2,5,7},{6}}
=> [[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,1,3,6,5,7,4] => ? = 1
{{1,3,4},{2,5},{6,7}}
=> [[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,1,3,6,7,5,4] => ? = 1
{{1,3,4},{2,5},{6},{7}}
=> [[1,3,4],[2,5],[6],[7]]
=> [7,6,2,5,1,3,4] => [2,1,3,7,6,5,4] => ? = 1
{{1,3,4,6,7},{2},{5}}
=> [[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000542
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 86%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
=> [[1,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
{{1},{2}}
=> [[1],[2]]
=> [2,1] => [2,1] => 2 = 1 + 1
{{1,2,3}}
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
{{1,2},{3}}
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 1 = 0 + 1
{{1,3},{2}}
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 2 + 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 1 = 0 + 1
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 1 = 0 + 1
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 1 = 0 + 1
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => 2 = 1 + 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => 2 = 1 + 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => 1 = 0 + 1
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => 1 = 0 + 1
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1 = 0 + 1
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => 2 = 1 + 1
{{1,2,5},{3},{4},{6},{7}}
=> [[1,2,5],[3],[4],[6],[7]]
=> [7,6,4,3,1,2,5] => [1,4,7,6,3,2,5] => ? = 0 + 1
{{1,2},{3,5},{4},{6},{7}}
=> [[1,2],[3,5],[4],[6],[7]]
=> [7,6,4,3,5,1,2] => [1,4,7,6,3,5,2] => ? = 0 + 1
{{1,2},{3},{4,5},{6},{7}}
=> [[1,2],[3,5],[4],[6],[7]]
=> [7,6,4,3,5,1,2] => [1,4,7,6,3,5,2] => ? = 0 + 1
{{1,2,6,7},{3},{4},{5}}
=> [[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,5,4,3,2,6,7] => ? = 0 + 1
{{1,2,6},{3,7},{4},{5}}
=> [[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,5,4,3,2,7,6] => ? = 0 + 1
{{1,2,6},{3},{4,7},{5}}
=> [[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,5,4,3,2,7,6] => ? = 0 + 1
{{1,2,6},{3},{4},{5,7}}
=> [[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,5,4,3,2,7,6] => ? = 0 + 1
{{1,2,6},{3},{4},{5},{7}}
=> [[1,2,6],[3],[4],[5],[7]]
=> [7,5,4,3,1,2,6] => [1,5,7,4,3,2,6] => ? = 0 + 1
{{1,2,7},{3,6},{4},{5}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0 + 1
{{1,2},{3,6,7},{4},{5}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0 + 1
{{1,2},{3,6},{4,7},{5}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0 + 1
{{1,2},{3,6},{4},{5,7}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0 + 1
{{1,2},{3,6},{4},{5},{7}}
=> [[1,2],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2] => [1,5,7,4,3,6,2] => ? = 0 + 1
{{1,2,7},{3},{4,6},{5}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0 + 1
{{1,2},{3,7},{4,6},{5}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0 + 1
{{1,2},{3},{4,6,7},{5}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0 + 1
{{1,2},{3},{4,6},{5,7}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0 + 1
{{1,2},{3},{4,6},{5},{7}}
=> [[1,2],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2] => [1,5,7,4,3,6,2] => ? = 0 + 1
{{1,2,7},{3},{4},{5,6}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0 + 1
{{1,2},{3,7},{4},{5,6}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0 + 1
{{1,2},{3},{4,7},{5,6}}
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,5,4,3,7,6,2] => ? = 0 + 1
{{1,2},{3},{4},{5,6,7}}
=> [[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,5,4,3,6,2,7] => ? = 0 + 1
{{1,2},{3},{4},{5,6},{7}}
=> [[1,2],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2] => [1,5,7,4,3,6,2] => ? = 0 + 1
{{1,2,7},{3},{4},{5},{6}}
=> [[1,2,7],[3],[4],[5],[6]]
=> [6,5,4,3,1,2,7] => [1,6,5,4,3,2,7] => ? = 0 + 1
{{1,2},{3,7},{4},{5},{6}}
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [1,6,5,4,3,7,2] => ? = 0 + 1
{{1,2},{3},{4,7},{5},{6}}
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [1,6,5,4,3,7,2] => ? = 0 + 1
{{1,2},{3},{4},{5,7},{6}}
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [1,6,5,4,3,7,2] => ? = 0 + 1
{{1,2},{3},{4},{5},{6,7}}
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [1,6,5,4,3,7,2] => ? = 0 + 1
{{1,2},{3},{4},{5},{6},{7}}
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => ? = 0 + 1
{{1,3,4,5,6,7},{2}}
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1 + 1
{{1,3,4,5,6},{2,7}}
=> [[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [2,1,3,4,5,7,6] => ? = 1 + 1
{{1,3,4,5,6},{2},{7}}
=> [[1,3,4,5,6],[2],[7]]
=> [7,2,1,3,4,5,6] => [2,1,3,4,7,5,6] => ? = 1 + 1
{{1,3,4,5,7},{2,6}}
=> [[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [2,1,3,4,6,5,7] => ? = 1 + 1
{{1,3,4,5},{2,6,7}}
=> [[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [2,1,3,4,6,7,5] => ? = 1 + 1
{{1,3,4,5},{2,6},{7}}
=> [[1,3,4,5],[2,6],[7]]
=> [7,2,6,1,3,4,5] => [2,1,3,4,7,6,5] => ? = 1 + 1
{{1,3,4,5,7},{2},{6}}
=> [[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [2,1,3,6,4,5,7] => ? = 1 + 1
{{1,3,4,5},{2,7},{6}}
=> [[1,3,4,5],[2,7],[6]]
=> [6,2,7,1,3,4,5] => [2,1,3,6,4,7,5] => ? = 1 + 1
{{1,3,4,5},{2},{6,7}}
=> [[1,3,4,5],[2,7],[6]]
=> [6,2,7,1,3,4,5] => [2,1,3,6,4,7,5] => ? = 1 + 1
{{1,3,4,5},{2},{6},{7}}
=> [[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [2,1,3,7,6,4,5] => ? = 1 + 1
{{1,3,4,6,7},{2,5}}
=> [[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => ? = 1 + 1
{{1,3,4,6},{2,5,7}}
=> [[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => ? = 1 + 1
{{1,3,4,6},{2,5},{7}}
=> [[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => ? = 1 + 1
{{1,3,4,7},{2,5,6}}
=> [[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [2,1,3,5,6,4,7] => ? = 1 + 1
{{1,3,4},{2,5,6,7}}
=> [[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [2,1,3,5,6,4,7] => ? = 1 + 1
{{1,3,4},{2,5,6},{7}}
=> [[1,3,4],[2,5,6],[7]]
=> [7,2,5,6,1,3,4] => [2,1,3,5,7,6,4] => ? = 1 + 1
{{1,3,4,7},{2,5},{6}}
=> [[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [2,1,3,6,5,4,7] => ? = 1 + 1
{{1,3,4},{2,5,7},{6}}
=> [[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,1,3,6,5,7,4] => ? = 1 + 1
{{1,3,4},{2,5},{6,7}}
=> [[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,1,3,6,7,5,4] => ? = 1 + 1
{{1,3,4},{2,5},{6},{7}}
=> [[1,3,4],[2,5],[6],[7]]
=> [7,6,2,5,1,3,4] => [2,1,3,7,6,5,4] => ? = 1 + 1
{{1,3,4,6,7},{2},{5}}
=> [[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 1 + 1
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000054
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [[1,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
{{1},{2}}
=> [[1],[2]]
=> [2,1] => [2,1] => 2 = 1 + 1
{{1,2,3}}
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
{{1,2},{3}}
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 1 = 0 + 1
{{1,3},{2}}
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1},{2,3}}
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1},{2},{3}}
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 2 + 1
{{1,2,3,4}}
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3},{4}}
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 1 = 0 + 1
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 1 = 0 + 1
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 1 = 0 + 1
{{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 1 = 0 + 1
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
{{1,3},{2,4}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => 2 = 1 + 1
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
{{1,4},{2,3}}
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => 2 = 1 + 1
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
{{1,2,3,4,5}}
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
{{1,2,3,4},{5}}
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1 = 0 + 1
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => 1 = 0 + 1
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1 = 0 + 1
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
{{1,2,4},{3,5}}
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => 1 = 0 + 1
{{1,2,5},{3,4}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1 = 0 + 1
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 1 = 0 + 1
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => 2 = 1 + 1
{{1,2,3,4,5},{6,7}}
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,2,3,4,6,7,5] => ? = 0 + 1
{{1,2,3,4,5},{6},{7}}
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => ? = 0 + 1
{{1,2,3,4,6,7},{5}}
=> [[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [1,2,3,5,4,6,7] => ? = 0 + 1
{{1,2,3,4,6},{5,7}}
=> [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [1,2,3,5,4,7,6] => ? = 0 + 1
{{1,2,3,4,6},{5},{7}}
=> [[1,2,3,4,6],[5],[7]]
=> [7,5,1,2,3,4,6] => [1,2,3,5,7,4,6] => ? = 0 + 1
{{1,2,3,4,7},{5,6}}
=> [[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [1,2,3,5,6,4,7] => ? = 0 + 1
{{1,2,3,4},{5,6,7}}
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,2,3,5,6,7,4] => ? = 0 + 1
{{1,2,3,4},{5,6},{7}}
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,3,5,7,6,4] => ? = 0 + 1
{{1,2,3,4,7},{5},{6}}
=> [[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [1,2,3,6,5,4,7] => ? = 0 + 1
{{1,2,3,4},{5,7},{6}}
=> [[1,2,3,4],[5,7],[6]]
=> [6,5,7,1,2,3,4] => [1,2,3,6,5,7,4] => ? = 0 + 1
{{1,2,3,4},{5},{6,7}}
=> [[1,2,3,4],[5,7],[6]]
=> [6,5,7,1,2,3,4] => [1,2,3,6,5,7,4] => ? = 0 + 1
{{1,2,3,4},{5},{6},{7}}
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => ? = 0 + 1
{{1,2,3,5,6,7},{4}}
=> [[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 0 + 1
{{1,2,3,5,6},{4,7}}
=> [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [1,2,4,3,5,7,6] => ? = 0 + 1
{{1,2,3,5,6},{4},{7}}
=> [[1,2,3,5,6],[4],[7]]
=> [7,4,1,2,3,5,6] => [1,2,4,3,7,5,6] => ? = 0 + 1
{{1,2,3,5,7},{4,6}}
=> [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [1,2,4,3,6,5,7] => ? = 0 + 1
{{1,2,3,5},{4,6,7}}
=> [[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [1,2,4,3,6,7,5] => ? = 0 + 1
{{1,2,3,5},{4,6},{7}}
=> [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => [1,2,4,3,7,6,5] => ? = 0 + 1
{{1,2,3,5,7},{4},{6}}
=> [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [1,2,4,6,3,5,7] => ? = 0 + 1
{{1,2,3,5},{4,7},{6}}
=> [[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [1,2,4,6,3,7,5] => ? = 0 + 1
{{1,2,3,5},{4},{6,7}}
=> [[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [1,2,4,6,3,7,5] => ? = 0 + 1
{{1,2,3,5},{4},{6},{7}}
=> [[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [1,2,4,7,6,3,5] => ? = 0 + 1
{{1,2,3,6,7},{4,5}}
=> [[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [1,2,4,5,3,6,7] => ? = 0 + 1
{{1,2,3,6},{4,5,7}}
=> [[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [1,2,4,5,3,7,6] => ? = 0 + 1
{{1,2,3,6},{4,5},{7}}
=> [[1,2,3,6],[4,5],[7]]
=> [7,4,5,1,2,3,6] => [1,2,4,5,7,3,6] => ? = 0 + 1
{{1,2,3,7},{4,5,6}}
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => ? = 0 + 1
{{1,2,3},{4,5,6,7}}
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => ? = 0 + 1
{{1,2,3},{4,5,6},{7}}
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [1,2,4,5,7,6,3] => ? = 0 + 1
{{1,2,3,7},{4,5},{6}}
=> [[1,2,3,7],[4,5],[6]]
=> [6,4,5,1,2,3,7] => [1,2,4,6,5,3,7] => ? = 0 + 1
{{1,2,3},{4,5,7},{6}}
=> [[1,2,3],[4,5,7],[6]]
=> [6,4,5,7,1,2,3] => [1,2,4,6,5,7,3] => ? = 0 + 1
{{1,2,3},{4,5},{6,7}}
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,2,4,6,7,5,3] => ? = 0 + 1
{{1,2,3},{4,5},{6},{7}}
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,2,4,7,6,5,3] => ? = 0 + 1
{{1,2,3,6,7},{4},{5}}
=> [[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [1,2,5,4,3,6,7] => ? = 0 + 1
{{1,2,3,6},{4,7},{5}}
=> [[1,2,3,6],[4,7],[5]]
=> [5,4,7,1,2,3,6] => [1,2,5,4,3,7,6] => ? = 0 + 1
{{1,2,3,6},{4},{5,7}}
=> [[1,2,3,6],[4,7],[5]]
=> [5,4,7,1,2,3,6] => [1,2,5,4,3,7,6] => ? = 0 + 1
{{1,2,3,6},{4},{5},{7}}
=> [[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [1,2,5,7,4,3,6] => ? = 0 + 1
{{1,2,3,7},{4,6},{5}}
=> [[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [1,2,5,4,6,3,7] => ? = 0 + 1
{{1,2,3},{4,6,7},{5}}
=> [[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [1,2,5,4,6,7,3] => ? = 0 + 1
{{1,2,3},{4,6},{5,7}}
=> [[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [1,2,5,4,7,6,3] => ? = 0 + 1
{{1,2,3},{4,6},{5},{7}}
=> [[1,2,3],[4,6],[5],[7]]
=> [7,5,4,6,1,2,3] => [1,2,5,7,4,6,3] => ? = 0 + 1
{{1,2,3,7},{4},{5,6}}
=> [[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [1,2,5,4,6,3,7] => ? = 0 + 1
{{1,2,3},{4,7},{5,6}}
=> [[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [1,2,5,4,7,6,3] => ? = 0 + 1
{{1,2,3},{4},{5,6,7}}
=> [[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [1,2,5,4,6,7,3] => ? = 0 + 1
{{1,2,3},{4},{5,6},{7}}
=> [[1,2,3],[4,6],[5],[7]]
=> [7,5,4,6,1,2,3] => [1,2,5,7,4,6,3] => ? = 0 + 1
{{1,2,3,7},{4},{5},{6}}
=> [[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [1,2,6,5,4,3,7] => ? = 0 + 1
{{1,2,3},{4,7},{5},{6}}
=> [[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [1,2,6,5,4,7,3] => ? = 0 + 1
{{1,2,3},{4},{5,7},{6}}
=> [[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [1,2,6,5,4,7,3] => ? = 0 + 1
{{1,2,3},{4},{5},{6,7}}
=> [[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [1,2,6,5,4,7,3] => ? = 0 + 1
{{1,2,3},{4},{5},{6},{7}}
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => ? = 0 + 1
{{1,2,4,5,6},{3,7}}
=> [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [1,3,2,4,5,7,6] => ? = 0 + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
The following 1 statistic also match your data. Click on any of them to see the details.
St000990The first ascent of a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!