Processing math: 70%

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000971
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => {{1}}
 => 1
{{1,2}}
 => [2,1] => [1,2] => {{1},{2}}
 => 1
{{1},{2}}
 => [1,2] => [2,1] => {{1,2}}
 => 2
{{1,2,3}}
 => [2,3,1] => [1,2,3] => {{1},{2},{3}}
 => 1
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => {{1},{2,3}}
 => 1
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => {{1,2},{3}}
 => 2
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => {{1,3},{2}}
 => 2
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => {{1,2,3}}
 => 3
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => {{1},{2,3,4}}
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => {{1,2},{3},{4}}
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => {{1,4},{2},{3}}
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => {{1,2},{3,4}}
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => {{1,3},{2},{4}}
 => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => {{1,2,3},{4}}
 => 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
 => 3
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
 => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,3,5,2] => {{1},{2,4,5},{3}}
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,3,5,4,2] => {{1},{2,3,5},{4}}
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
 => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
 => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,2,5,3] => {{1,4,5},{2},{3}}
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,1,4,2,3] => {{1,5},{2},{3,4}}
 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
 => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
 => 2
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: St000326
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => => ? = 1
{{1,2}}
 => [2,1] => [2,1] => 1 => 1
{{1},{2}}
 => [1,2] => [1,2] => 0 => 2
{{1,2,3}}
 => [2,3,1] => [3,2,1] => 11 => 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 10 => 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => 01 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 01 => 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 00 => 3
{{1,2,3,4}}
 => [2,3,4,1] => [4,3,2,1] => 111 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,2,1,4] => 110 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,2,1] => 101 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 101 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => 100 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,3,1] => 011 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => 011 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => 010 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => 011 => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => 011 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => 010 => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => 001 => 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => 001 => 3
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => 001 => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 000 => 4
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,4,3,2,1] => 1111 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,3,2,1,5] => 1110 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,3,2,1] => 1101 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,2,1,5,4] => 1101 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,2,1,4,5] => 1100 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,4,2,1] => 1011 => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,4,3] => 1011 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,2,1,5] => 1010 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,2,1] => 1011 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,2,1] => 1001 => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => 1001 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,4,3,1] => 0111 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,5,3,2] => 0111 => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,3,1,5] => 0110 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,2,4,3,1] => 0111 => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,4,1,3,2] => 0111 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,3,2,5] => 0110 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,3,1] => 0101 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,5,1,3,2] => 0101 => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => 0101 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => 0100 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,4,1] => 0111 => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,3,1,4,2] => 0111 => 2
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [3,2,4,1,5] => 0110 => 2
{{1},{2,4,7},{3},{5,8},{6}}
 => [1,4,3,7,8,6,2,5] => [1,3,6,8,4,2,7,5] => ? => ? = 3
{{1,7},{2,5},{3,6},{4},{8}}
 => [7,5,6,4,2,3,1,8] => [4,6,2,5,3,7,1,8] => ? => ? = 4
{{1,3,5,7,9},{2},{4},{6},{8},{10}}
 => [3,2,5,4,7,6,9,8,1,10] => [2,4,6,8,9,7,5,3,1,10] => ? => ? = 2
{{1,3,5,7,9,11},{2},{4},{6},{8},{10},{12}}
 => [3,2,5,4,7,6,9,8,11,10,1,12] => [2,4,6,8,10,11,9,7,5,3,1,12] => ? => ? = 2
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,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000745
(load all 64 compositions to match this statistic)
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 78% ●values known / values provided: 98%●distinct values known / distinct values provided: 78%
Values
{{1}}
 => [[1]]
 => 1
{{1,2}}
 => [[1,2]]
 => 1
{{1},{2}}
 => [[1],[2]]
 => 2
{{1,2,3}}
 => [[1,2,3]]
 => 1
{{1,2},{3}}
 => [[1,2],[3]]
 => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => 2
{{1},{2,3}}
 => [[1,3],[2]]
 => 2
{{1},{2},{3}}
 => [[1],[2],[3]]
 => 3
{{1,2,3,4}}
 => [[1,2,3,4]]
 => 1
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 1
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 2
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => 2
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 2
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => 2
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 2
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 2
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 3
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 3
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 3
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 4
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => 1
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => 1
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => 1
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => 1
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => 1
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => 1
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => 1
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => 1
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => 1
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => 1
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => 1
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => 1
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => 1
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => 1
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => 2
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => 2
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => 2
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => 2
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => 2
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => 2
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => 2
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => 2
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => 2
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => 2
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => 2
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => 2
{{1},{2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? = 8
{{1},{2},{3},{4},{5},{6},{7,8}}
 => ?
 => ? = 7
{{1,2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? = 1
{{1,2},{3},{4},{5},{6},{7,8}}
 => ?
 => ? = 1
{{1,2},{3,4},{5},{6},{7},{8}}
 => ?
 => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8}}
 => ?
 => ? = 2
{{1,7},{2},{3},{4},{5},{6},{8}}
 => ?
 => ? = 6
{{1,8},{2},{3},{4},{5},{6},{7}}
 => ?
 => ? = 7
{{1,8},{2},{3},{4},{5},{6,7}}
 => ?
 => ? = 6
{{1,2,3},{4},{5},{6},{7},{8}}
 => ?
 => ? = 1
{{1,4,7},{2,6},{3},{5},{8}}
 => ?
 => ? = 3
{{1,3},{2,4},{5},{6},{7},{8}}
 => ?
 => ? = 2
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? = 8
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ?
 => ? = 9
{{1},{2,4,7},{3},{5,8},{6}}
 => ?
 => ? = 3
{{1},{2,5,8},{3,7},{4},{6}}
 => ?
 => ? = 4
{{1},{2,6},{3,7},{4,8},{5}}
 => ?
 => ? = 5
{{1},{2,5,7},{3,8},{4},{6}}
 => ?
 => ? = 4
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
 => ?
 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => ?
 => ? = 1
{{1,6},{2,4,8},{3},{5},{7}}
 => ?
 => ? = 3
{{1},{2,10},{3,9},{4,8},{5,7},{6}}
 => ?
 => ? = 6
{{1,7},{2,5},{3,6},{4},{8}}
 => ?
 => ? = 4
{{1,3,5,7,9},{2},{4},{6},{8},{10}}
 => ?
 => ? = 2
{{1,3,5,7,9,11},{2},{4},{6},{8},{10},{12}}
 => ?
 => ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
Mapping
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: 78% ●values known / values provided: 98%●distinct values known / distinct values provided: 78%
Values
{{1}}
 => [[1]]
 => [[1]]
 => [1] => 1
{{1,2}}
 => [[1,2]]
 => [[1],[2]]
 => [1,1] => 1
{{1},{2}}
 => [[1],[2]]
 => [[1,2]]
 => [2] => 2
{{1,2,3}}
 => [[1,2,3]]
 => [[1],[2],[3]]
 => [1,1,1] => 1
{{1,2},{3}}
 => [[1,2],[3]]
 => [[1,3],[2]]
 => [1,2] => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => [[1,2],[3]]
 => [2,1] => 2
{{1},{2,3}}
 => [[1,3],[2]]
 => [[1,2],[3]]
 => [2,1] => 2
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [[1,2,3]]
 => [3] => 3
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => [1,2,1] => 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => [1,2,1] => 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => [1,3] => 1
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => [2,2] => 2
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => 2
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => [2,2] => 2
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [2,2] => 2
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => [3,1] => 3
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => [3,1] => 3
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => [3,1] => 3
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => [4] => 4
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => [1,1,2,1] => 1
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => [1,1,2,1] => 1
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => [1,2,1,1] => 1
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => [1,2,2] => 1
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => [1,2,1,1] => 1
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => [1,2,1,1] => 1
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => [1,2,2] => 1
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => [1,3,1] => 1
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => [1,3,1] => 1
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => [1,3,1] => 1
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => [2,1,2] => 2
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => [2,1,2] => 2
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => [2,3] => 2
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [2,2,1] => 2
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => [2,2,1] => 2
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => [2,2,1] => 2
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => [2,3] => 2
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
{{1},{2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ?
 => ? => ? = 8
{{1},{2},{3},{4},{5},{6},{7,8}}
 => ?
 => ?
 => ? => ? = 7
{{1,2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ?
 => ? => ? = 1
{{1,2},{3},{4},{5},{6},{7,8}}
 => ?
 => ?
 => ? => ? = 1
{{1,2},{3,4},{5},{6},{7},{8}}
 => ?
 => ?
 => ? => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8}}
 => ?
 => ?
 => ? => ? = 2
{{1,7},{2},{3},{4},{5},{6},{8}}
 => ?
 => ?
 => ? => ? = 6
{{1,8},{2},{3},{4},{5},{6},{7}}
 => ?
 => ?
 => ? => ? = 7
{{1,8},{2},{3},{4},{5},{6,7}}
 => ?
 => ?
 => ? => ? = 6
{{1,2,3},{4},{5},{6},{7},{8}}
 => ?
 => ?
 => ? => ? = 1
{{1,4,7},{2,6},{3},{5},{8}}
 => ?
 => ?
 => ? => ? = 3
{{1,3},{2,4},{5},{6},{7},{8}}
 => ?
 => ?
 => ? => ? = 2
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ?
 => ? => ? = 8
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ?
 => ?
 => ? => ? = 9
{{1},{2,4,7},{3},{5,8},{6}}
 => ?
 => ?
 => ? => ? = 3
{{1},{2,5,8},{3,7},{4},{6}}
 => ?
 => ?
 => ? => ? = 4
{{1},{2,6},{3,7},{4,8},{5}}
 => ?
 => ?
 => ? => ? = 5
{{1},{2,5,7},{3,8},{4},{6}}
 => ?
 => ?
 => ? => ? = 4
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
 => ?
 => ?
 => ? => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => ?
 => ?
 => ? => ? = 1
{{1,6},{2,4,8},{3},{5},{7}}
 => ?
 => ?
 => ? => ? = 3
{{1},{2,10},{3,9},{4,8},{5,7},{6}}
 => ?
 => ?
 => ? => ? = 6
{{1,7},{2,5},{3,6},{4},{8}}
 => ?
 => ?
 => ? => ? = 4
{{1,3,5,7,9},{2},{4},{6},{8},{10}}
 => ?
 => ?
 => ? => ? = 2
{{1,3,5,7,9,11},{2},{4},{6},{8},{10},{12}}
 => ?
 => ?
 => ? => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000297
(load all 7 compositions to match this statistic)
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 98%●distinct values known / distinct values provided: 78%
Values
{{1}}
 => [[1]]
 => => ? = 1 - 1
{{1,2}}
 => [[1,2]]
 => 0 => 0 = 1 - 1
{{1},{2}}
 => [[1],[2]]
 => 1 => 1 = 2 - 1
{{1,2,3}}
 => [[1,2,3]]
 => 00 => 0 = 1 - 1
{{1,2},{3}}
 => [[1,2],[3]]
 => 01 => 0 = 1 - 1
{{1,3},{2}}
 => [[1,3],[2]]
 => 10 => 1 = 2 - 1
{{1},{2,3}}
 => [[1,3],[2]]
 => 10 => 1 = 2 - 1
{{1},{2},{3}}
 => [[1],[2],[3]]
 => 11 => 2 = 3 - 1
{{1,2,3,4}}
 => [[1,2,3,4]]
 => 000 => 0 = 1 - 1
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 001 => 0 = 1 - 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 010 => 0 = 1 - 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => 010 => 0 = 1 - 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 011 => 0 = 1 - 1
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 100 => 1 = 2 - 1
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => 101 => 1 = 2 - 1
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 101 => 1 = 2 - 1
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => 101 => 1 = 2 - 1
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 100 => 1 = 2 - 1
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 101 => 1 = 2 - 1
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 110 => 2 = 3 - 1
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 110 => 2 = 3 - 1
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 110 => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 111 => 3 = 4 - 1
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => 0000 => 0 = 1 - 1
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => 0001 => 0 = 1 - 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => 0010 => 0 = 1 - 1
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => 0010 => 0 = 1 - 1
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => 0011 => 0 = 1 - 1
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => 0100 => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => 0101 => 0 = 1 - 1
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => 0101 => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => 0100 => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => 0100 => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => 0101 => 0 = 1 - 1
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => 0110 => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => 0110 => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => 0110 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => 0111 => 0 = 1 - 1
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => 1000 => 1 = 2 - 1
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => 1001 => 1 = 2 - 1
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => 1001 => 1 = 2 - 1
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => 1010 => 1 = 2 - 1
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => 1010 => 1 = 2 - 1
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => 1011 => 1 = 2 - 1
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => 1010 => 1 = 2 - 1
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => 1010 => 1 = 2 - 1
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => 1010 => 1 = 2 - 1
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => 1011 => 1 = 2 - 1
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => 1010 => 1 = 2 - 1
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => 1010 => 1 = 2 - 1
{{1,4},{2,3},{5}}
 => [[1,3],[2,4],[5]]
 => 1011 => 1 = 2 - 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? = 8 - 1
{{1},{2},{3},{4},{5},{6},{7,8}}
 => ?
 => ? => ? = 7 - 1
{{1,2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? = 1 - 1
{{1,2},{3},{4},{5},{6},{7,8}}
 => ?
 => ? => ? = 1 - 1
{{1,2},{3,4},{5},{6},{7},{8}}
 => ?
 => ? => ? = 1 - 1
{{1,3},{2},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? = 2 - 1
{{1,7},{2},{3},{4},{5},{6},{8}}
 => ?
 => ? => ? = 6 - 1
{{1,8},{2},{3},{4},{5},{6},{7}}
 => ?
 => ? => ? = 7 - 1
{{1,8},{2},{3},{4},{5},{6,7}}
 => ?
 => ? => ? = 6 - 1
{{1,2,3},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? = 1 - 1
{{1,4,7},{2,6},{3},{5},{8}}
 => ?
 => ? => ? = 3 - 1
{{1,3},{2,4},{5},{6},{7},{8}}
 => ?
 => ? => ? = 2 - 1
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? = 8 - 1
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ?
 => ? => ? = 9 - 1
{{1},{2,4,7},{3},{5,8},{6}}
 => ?
 => ? => ? = 3 - 1
{{1},{2,5,8},{3,7},{4},{6}}
 => ?
 => ? => ? = 4 - 1
{{1},{2,6},{3,7},{4,8},{5}}
 => ?
 => ? => ? = 5 - 1
{{1},{2,5,7},{3,8},{4},{6}}
 => ?
 => ? => ? = 4 - 1
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
 => ?
 => ? => ? = 1 - 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => ?
 => ? => ? = 1 - 1
{{1,6},{2,4,8},{3},{5},{7}}
 => ?
 => ? => ? = 3 - 1
{{1},{2,10},{3,9},{4,8},{5,7},{6}}
 => ?
 => ? => ? = 6 - 1
{{1,7},{2,5},{3,6},{4},{8}}
 => ?
 => ? => ? = 4 - 1
{{1,3,5,7,9},{2},{4},{6},{8},{10}}
 => ?
 => ? => ? = 2 - 1
{{1,3,5,7,9,11},{2},{4},{6},{8},{10},{12}}
 => ?
 => ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000383
Mapping
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: 78% ●values known / values provided: 98%●distinct values known / distinct values provided: 78%
Values
{{1}}
 => [[1]]
 => [1] => [1] => 1
{{1,2}}
 => [[1,2]]
 => [2] => [1,1] => 1
{{1},{2}}
 => [[1],[2]]
 => [1,1] => [2] => 2
{{1,2,3}}
 => [[1,2,3]]
 => [3] => [1,1,1] => 1
{{1,2},{3}}
 => [[1,2],[3]]
 => [2,1] => [2,1] => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => [1,2] => [1,2] => 2
{{1},{2,3}}
 => [[1,3],[2]]
 => [1,2] => [1,2] => 2
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [1,1,1] => [3] => 3
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [3,1] => [2,1,1] => 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [2,2] => [1,2,1] => 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [2,2] => [1,2,1] => 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [2,1,1] => [3,1] => 1
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [1,3] => [1,1,2] => 2
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [1,2,1] => [2,2] => 2
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [1,2,1] => [2,2] => 2
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [1,2,1] => [2,2] => 2
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [1,3] => [1,1,2] => 2
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [1,2,1] => [2,2] => 2
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [1,1,2] => [1,3] => 3
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [1,1,2] => [1,3] => 3
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [1,1,2] => [1,3] => 3
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [4,1] => [2,1,1,1] => 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [3,2] => [1,2,1,1] => 1
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [3,2] => [1,2,1,1] => 1
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [3,1,1] => [3,1,1] => 1
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [2,3] => [1,1,2,1] => 1
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [2,2,1] => [2,2,1] => 1
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [2,2,1] => [2,2,1] => 1
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [2,3] => [1,1,2,1] => 1
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [2,3] => [1,1,2,1] => 1
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [2,2,1] => [2,2,1] => 1
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => [4,1] => 1
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => [1,4] => [1,1,1,2] => 2
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => [1,3,1] => [2,1,2] => 2
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => [1,3,1] => [2,1,2] => 2
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [1,2,2] => [1,2,2] => 2
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [1,2,2] => [1,2,2] => 2
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [1,2,1,1] => [3,2] => 2
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => [1,2,2] => [1,2,2] => 2
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => [1,2,2] => [1,2,2] => 2
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => [1,2,2] => [1,2,2] => 2
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [1,2,1,1] => [3,2] => 2
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [1,2,2] => [1,2,2] => 2
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [1,2,2] => [1,2,2] => 2
{{1},{2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? => ? = 8
{{1},{2},{3},{4},{5},{6},{7,8}}
 => ?
 => ? => ? => ? = 7
{{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] => ? = 4
{{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] => ? = 4
{{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] => ? = 5
{{1,2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? => ? = 1
{{1,2},{3},{4},{5},{6},{7,8}}
 => ?
 => ? => ? => ? = 1
{{1,2},{3,4},{5},{6},{7},{8}}
 => ?
 => ? => ? => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? => ? = 2
{{1,7},{2},{3},{4},{5},{6},{8}}
 => ?
 => ? => ? => ? = 6
{{1,8},{2},{3},{4},{5},{6},{7}}
 => ?
 => ? => ? => ? = 7
{{1,8},{2},{3},{4},{5},{6,7}}
 => ?
 => ? => ? => ? = 6
{{1,2,3},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? => ? = 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] => ? = 4
{{1,4,7},{2,6},{3},{5},{8}}
 => ?
 => ? => ? => ? = 3
{{1,3},{2,4},{5},{6},{7},{8}}
 => ?
 => ? => ? => ? = 2
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ?
 => ? => ? => ? = 8
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ?
 => ? => ? => ? = 9
{{1},{2,4,7},{3},{5,8},{6}}
 => ?
 => ? => ? => ? = 3
{{1},{2,5,8},{3,7},{4},{6}}
 => ?
 => ? => ? => ? = 4
{{1},{2,6},{3,7},{4,8},{5}}
 => ?
 => ? => ? => ? = 5
{{1},{2,5,7},{3,8},{4},{6}}
 => ?
 => ? => ? => ? = 4
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
 => ?
 => ? => ? => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => ?
 => ? => ? => ? = 1
{{1,6},{2,4,8},{3},{5},{7}}
 => ?
 => ? => ? => ? = 3
{{1},{2,10},{3,9},{4,8},{5,7},{6}}
 => ?
 => ? => ? => ? = 6
{{1,7},{2,5},{3,6},{4},{8}}
 => ?
 => ? => ? => ? = 4
{{1,3,5,7,9},{2},{4},{6},{8},{10}}
 => ?
 => ? => ? => ? = 2
{{1,3,5,7,9,11},{2},{4},{6},{8},{10},{12}}
 => ?
 => ? => ? => ? = 2
Description
The last part of an integer composition.
Matching statistic: St001050
(load all 217 compositions to match this statistic)
Mapping
Mp00112: Set partitions —complement⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 78% ●values known / values provided: 97%●distinct values known / distinct values provided: 78%
Values
{{1}}
 => {{1}}
 => 1
{{1,2}}
 => {{1,2}}
 => 1
{{1},{2}}
 => {{1},{2}}
 => 2
{{1,2,3}}
 => {{1,2,3}}
 => 1
{{1,2},{3}}
 => {{1},{2,3}}
 => 1
{{1,3},{2}}
 => {{1,3},{2}}
 => 2
{{1},{2,3}}
 => {{1,2},{3}}
 => 2
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 3
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 1
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 1
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 1
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 2
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => 2
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 2
{{1,4},{2,3}}
 => {{1,4},{2,3}}
 => 2
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 2
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 2
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 3
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 3
{{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 3
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 4
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 1
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => 1
{{1,2,3,5},{4}}
 => {{1,3,4,5},{2}}
 => 1
{{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => 1
{{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => 1
{{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => 1
{{1,2,4},{3,5}}
 => {{1,3},{2,4,5}}
 => 1
{{1,2,4},{3},{5}}
 => {{1},{2,4,5},{3}}
 => 1
{{1,2,5},{3,4}}
 => {{1,4,5},{2,3}}
 => 1
{{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => 1
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => 1
{{1,2,5},{3},{4}}
 => {{1,4,5},{2},{3}}
 => 1
{{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => 1
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => 1
{{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => 1
{{1,3,4,5},{2}}
 => {{1,2,3,5},{4}}
 => 2
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => 2
{{1,3,5},{2,4}}
 => {{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => {{1,2,4},{3,5}}
 => 2
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => 2
{{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => 2
{{1,3},{2,5},{4}}
 => {{1,4},{2},{3,5}}
 => 2
{{1,3},{2},{4,5}}
 => {{1,2},{3,5},{4}}
 => 2
{{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => 2
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => 2
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => 2
{{1},{2},{3},{4},{5},{6},{7},{8}}
 => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 8
{{1},{2},{3},{4},{5},{6},{7,8}}
 => {{1,2},{3},{4},{5},{6},{7},{8}}
 => ? = 7
{{1},{2,4,6,8},{3},{5},{7}}
 => {{1,3,5,7},{2},{4},{6},{8}}
 => ? = 3
{{1},{2,4,8},{3},{5,7},{6}}
 => {{1,5,7},{2,4},{3},{6},{8}}
 => ? = 3
{{1},{2,6,8},{3,5},{4},{7}}
 => {{1,3,7},{2},{4,6},{5},{8}}
 => ? = 4
{{1},{2,8},{3,5,7},{4},{6}}
 => {{1,7},{2,4,6},{3},{5},{8}}
 => ? = 4
{{1},{2,8},{3,7},{4,6},{5}}
 => {{1,7},{2,6},{3,5},{4},{8}}
 => ? = 5
{{1,2},{3},{4},{5},{6},{7},{8}}
 => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 1
{{1,2},{3},{4},{5},{6},{7,8}}
 => {{1,2},{3},{4},{5},{6},{7,8}}
 => ? = 1
{{1,2},{3,4},{5},{6},{7},{8}}
 => {{1},{2},{3},{4},{5,6},{7,8}}
 => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8}}
 => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 2
{{1,3},{2},{4,6,8},{5},{7}}
 => {{1,3,5},{2},{4},{6,8},{7}}
 => ? = 2
{{1,3},{2},{4,8},{5,7},{6}}
 => {{1,5},{2,4},{3},{6,8},{7}}
 => ? = 2
{{1,7},{2},{3},{4},{5},{6},{8}}
 => {{1},{2,8},{3},{4},{5},{6},{7}}
 => ? = 6
{{1,8},{2},{3},{4},{5},{6},{7}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 7
{{1,8},{2},{3},{4},{5},{6,7}}
 => {{1,8},{2,3},{4},{5},{6},{7}}
 => ? = 6
{{1,3,5},{2},{4},{6,8},{7}}
 => {{1,3},{2},{4,6,8},{5},{7}}
 => ? = 2
{{1,3,5,7},{2},{4},{6},{8}}
 => {{1},{2,4,6,8},{3},{5},{7}}
 => ? = 2
{{1,3,7},{2},{4,6},{5},{8}}
 => {{1},{2,6,8},{3,5},{4},{7}}
 => ? = 2
{{1,2,3},{4},{5},{6},{7},{8}}
 => {{1},{2},{3},{4},{5},{6,7,8}}
 => ? = 1
{{1,5},{2,4},{3},{6,8},{7}}
 => {{1,3},{2},{4,8},{5,7},{6}}
 => ? = 3
{{1,5,7},{2,4},{3},{6},{8}}
 => {{1},{2,4,8},{3},{5,7},{6}}
 => ? = 3
{{1,7},{2,4,6},{3},{5},{8}}
 => {{1},{2,8},{3,5,7},{4},{6}}
 => ? = 3
{{1,7},{2,6},{3,5},{4},{8}}
 => {{1},{2,8},{3,7},{4,6},{5}}
 => ? = 4
{{1,4,7},{2,6},{3},{5},{8}}
 => {{1},{2,5,8},{3,7},{4},{6}}
 => ? = 3
{{1,3},{2,4},{5},{6},{7},{8}}
 => {{1},{2},{3},{4},{5,7},{6,8}}
 => ? = 2
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 8
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => ? = 9
{{1},{2,4,7},{3},{5,8},{6}}
 => {{1,4},{2,5,7},{3},{6},{8}}
 => ? = 3
{{1},{2,5,8},{3,7},{4},{6}}
 => {{1,4,7},{2,6},{3},{5},{8}}
 => ? = 4
{{1},{2,6},{3,7},{4,8},{5}}
 => {{1,5},{2,6},{3,7},{4},{8}}
 => ? = 5
{{1},{2,5,7},{3,8},{4},{6}}
 => {{1,6},{2,4,7},{3},{5},{8}}
 => ? = 4
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
 => {{1},{2},{3},{4},{5},{6},{7},{8,9}}
 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => {{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
 => ? = 1
{{1,6},{2,4,8},{3},{5},{7}}
 => {{1,5,7},{2},{3,8},{4},{6}}
 => ? = 3
{{1},{2,10},{3,9},{4,8},{5,7},{6}}
 => {{1,9},{2,8},{3,7},{4,6},{5},{10}}
 => ? = 6
{{1,7},{2,5},{3,6},{4},{8}}
 => {{1},{2,8},{3,6},{4,7},{5}}
 => ? = 4
{{1,3,5,7,9},{2},{4},{6},{8},{10}}
 => {{1},{2,4,6,8,10},{3},{5},{7},{9}}
 => ? = 2
{{1,3,5,7,9,11},{2},{4},{6},{8},{10},{12}}
 => {{1},{2,4,6,8,10,12},{3},{5},{7},{9},{11}}
 => ? = 2
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: St000542
Mapping
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: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [[1]]
 => [1] => [1] => 1
{{1,2}}
 => [[1,2]]
 => [1,2] => [1,2] => 1
{{1},{2}}
 => [[1],[2]]
 => [2,1] => [2,1] => 2
{{1,2,3}}
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 1
{{1,2},{3}}
 => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 2
{{1},{2,3}}
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 2
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 3
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 1
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [3,1,2,4] => [1,3,2,4] => 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [3,4,1,2] => [1,3,4,2] => 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => 1
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 2
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => [2,1,4,3] => 2
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => 2
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => [2,1,4,3] => 2
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 2
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => 2
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 3
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 3
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 3
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 4
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,4,3,5] => 1
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,4,5,3] => 1
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 1
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,3,2,4,5] => 1
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,3,2,5,4] => 1
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,3,5,2,4] => 1
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,3,4,2,5] => 1
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,3,4,2,5] => 1
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,3,5,4,2] => 1
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,4,3,2,5] => 1
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,4,3,5,2] => 1
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,4,3,5,2] => 1
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,5,4,3,2] => 1
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 2
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,1,3,5,4] => 2
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,1,5,3,4] => 2
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 2
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 2
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [2,1,5,4,3] => 2
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [2,4,1,3,5] => 2
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => 2
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => 2
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [2,5,4,1,3] => 2
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 2
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 2
{{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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 2
Description
The number of left-to-right-minima of a permutation. An integer σi in the one-line notation of a permutation σ is a left-to-right-minimum if there does not exist a j < i such that σj<σi.
Matching statistic: St000541
Mapping
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: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [[1]]
 => [1] => [1] => ? = 1 - 1
{{1,2}}
 => [[1,2]]
 => [1,2] => [1,2] => 0 = 1 - 1
{{1},{2}}
 => [[1],[2]]
 => [2,1] => [2,1] => 1 = 2 - 1
{{1,2,3}}
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2},{3}}
 => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 0 = 1 - 1
{{1,3},{2}}
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1},{2,3}}
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [3,1,2,4] => [1,3,2,4] => 0 = 1 - 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [3,4,1,2] => [1,3,4,2] => 0 = 1 - 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => 0 = 1 - 1
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => [2,1,4,3] => 1 = 2 - 1
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => [2,1,4,3] => 1 = 2 - 1
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,4,3,5] => 0 = 1 - 1
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,4,5,3] => 0 = 1 - 1
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 0 = 1 - 1
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,3,2,5,4] => 0 = 1 - 1
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,3,5,2,4] => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,3,4,2,5] => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,3,4,2,5] => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,3,5,4,2] => 0 = 1 - 1
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,4,3,2,5] => 0 = 1 - 1
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,4,3,5,2] => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,4,3,5,2] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,5,4,3,2] => 0 = 1 - 1
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,1,3,5,4] => 1 = 2 - 1
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,1,5,3,4] => 1 = 2 - 1
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 1 = 2 - 1
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 1 = 2 - 1
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [2,1,5,4,3] => 1 = 2 - 1
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [2,4,1,3,5] => 1 = 2 - 1
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => 1 = 2 - 1
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => 1 = 2 - 1
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [2,5,4,1,3] => 1 = 2 - 1
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 1 = 2 - 1
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 1 = 2 - 1
{{1,4},{2,3},{5}}
 => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [2,1,5,4,3] => 1 = 2 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 1 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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] => ? = 2 - 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 π of length n, this is the number of indices 2≤j≤n such that for all 1≤i<j, the pair (i,j) is an inversion of π.
Matching statistic: St000054
Mapping
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: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 78%
Values
{{1}}
 => [[1]]
 => [1] => [1] => 1
{{1,2}}
 => [[1,2]]
 => [1,2] => [1,2] => 1
{{1},{2}}
 => [[1],[2]]
 => [2,1] => [2,1] => 2
{{1,2,3}}
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 1
{{1,2},{3}}
 => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 2
{{1},{2,3}}
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 2
{{1},{2},{3}}
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 3
{{1,2,3,4}}
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 1
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => [3,1,2,4] => [1,3,2,4] => 1
{{1,2},{3,4}}
 => [[1,2],[3,4]]
 => [3,4,1,2] => [1,3,4,2] => 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => 1
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 2
{{1,3},{2,4}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => [2,1,4,3] => 2
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => 2
{{1,4},{2,3}}
 => [[1,3],[2,4]]
 => [2,4,1,3] => [2,1,4,3] => 2
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 2
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => 2
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 3
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 3
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 3
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 4
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,4,3,5] => 1
{{1,2,3},{4,5}}
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,4,5,3] => 1
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 1
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,3,2,4,5] => 1
{{1,2,4},{3,5}}
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,3,2,5,4] => 1
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,3,5,2,4] => 1
{{1,2,5},{3,4}}
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,3,4,2,5] => 1
{{1,2},{3,4,5}}
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,3,4,2,5] => 1
{{1,2},{3,4},{5}}
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,3,5,4,2] => 1
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,4,3,2,5] => 1
{{1,2},{3,5},{4}}
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,4,3,5,2] => 1
{{1,2},{3},{4,5}}
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,4,3,5,2] => 1
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,5,4,3,2] => 1
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 2
{{1,3,4},{2,5}}
 => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,1,3,5,4] => 2
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,1,5,3,4] => 2
{{1,3,5},{2,4}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 2
{{1,3},{2,4,5}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 2
{{1,3},{2,4},{5}}
 => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [2,1,5,4,3] => 2
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [2,4,1,3,5] => 2
{{1,3},{2,5},{4}}
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => 2
{{1,3},{2},{4,5}}
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => 2
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [2,5,4,1,3] => 2
{{1,4,5},{2,3}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 2
{{1,4},{2,3,5}}
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [2,1,4,3,5] => 2
{{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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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 π of n, together with its rotations, obtained by conjugating with the long cycle (1,…,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. 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.