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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000941
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St001200
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 3
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 3
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 3
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 3
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 3
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 3
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 3
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,4,6],[3],[5],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,3,6],[4],[5],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,3,4,5],[2],[6],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,4,5],[3],[6],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,3,5],[4],[6],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0 + 3
[[1,5,7],[2,6],[3],[4]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,4,7],[2,6],[3],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,7],[2,6],[4],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,7],[3,6],[4],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,4,7],[2,5],[3],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,7],[2,5],[4],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,7],[3,5],[4],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,7],[2,4],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,7],[3,4],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,5,6],[2,7],[3],[4]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,4,6],[2,7],[3],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,6],[2,7],[4],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,6],[3,7],[4],[5]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,4,5],[2,7],[3],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,5],[2,7],[4],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,5],[3,7],[4],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,4],[2,7],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,4],[3,7],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,3],[4,7],[5],[6]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,4,6],[2,5],[3],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,6],[2,5],[4],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,6],[3,5],[4],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,6],[2,4],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,6],[3,4],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,4,5],[2,6],[3],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,5],[2,6],[4],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,5],[3,6],[4],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,4],[2,6],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,4],[3,6],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,3],[4,6],[5],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,5],[2,4],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,5],[3,4],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,3,4],[2,5],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,4],[3,5],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000370
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000370: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000370: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => ([(0,6),(1,3),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => ([(0,6),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => ([(0,3),(0,4),(0,6),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => ([(0,2),(0,3),(0,4),(0,6),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
Description
The genus of a graph.
This is the smallest genus of an oriented surface on which the graph can be embedded without crossings. One can indeed compute the genus as the sum of the genuses for the connected components.
Matching statistic: St001871
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001871: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001871: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [2,5] => ([(4,6),(5,6)],7)
=> ? = 0
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [2,5] => ([(4,6),(5,6)],7)
=> ? = 0
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
Description
The number of triconnected components of a graph.
A connected graph is '''triconnected''' or '''3-vertex connected''' if it cannot be disconnected by removing two or fewer vertices. An arbitrary connected graph can be decomposed as a union of biconnected (2-vertex connected) graphs, known as '''blocks''', and each biconnected graph can be decomposed as a union of components with are either a cycle (type "S"), a cocyle (type "P"), or triconnected (type "R"). The decomposition of a biconnected graph into these components is known as the '''SPQR-tree''' of the graph.
Matching statistic: St001728
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001728: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001728: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1 = 0 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1 = 0 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 2 = 1 + 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1 = 0 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 1 = 0 + 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 1 = 0 + 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 1 = 0 + 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 1 = 0 + 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 1 = 0 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2 = 1 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 2 = 1 + 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0 + 1
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 0 + 1
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
Description
The number of invisible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$. Thus, an invisible descent satisfies $\pi(i) > \pi(i+1) > i$.
Matching statistic: St001345
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001345: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001345: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => ([(0,6),(1,3),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => ([(0,6),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => ([(0,3),(0,4),(0,6),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => ([(0,2),(0,3),(0,4),(0,6),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
Description
The Hamming dimension of a graph.
Let $H(n, k)$ be the graph whose vertices are the subsets of $\{1,\dots,n\}$, and $(u,v)$ being an edge, for $u\neq v$, if the symmetric difference of $u$ and $v$ has cardinality at most $k$.
This statistic is the smallest $n$ such that the graph is an induced subgraph of $H(n, k)$ for some $k$.
Matching statistic: St001716
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001716: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001716: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => ([(0,6),(1,3),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => ([(0,6),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => ([(0,3),(0,4),(0,6),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => ([(0,2),(0,3),(0,4),(0,6),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
Description
The 1-improper chromatic number of a graph.
This is the least number of colours in a vertex-colouring, such that each vertex has at most one neighbour with the same colour.
Matching statistic: St001792
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001792: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001792: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => ([(0,6),(1,3),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => ([(0,6),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => ([(0,3),(0,4),(0,6),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => ([(0,2),(0,3),(0,4),(0,6),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 0 + 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => ([(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0 + 2
Description
The arboricity of a graph.
This is the minimum number of forests that covers all edges of the graph.
Matching statistic: St000455
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [2,5] => ([(4,6),(5,6)],7)
=> 0
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [2,5] => ([(4,6),(5,6)],7)
=> 0
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[[1,2,3],[4,6],[5],[7]]
=> [7,5,4,6,1,2,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001331
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001331: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001331: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 1 + 3
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 3
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 3
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 3
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 3
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 3
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 3
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 3
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ([(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => ([(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,2),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => ([(0,6),(1,3),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,3),(1,6),(1,7),(2,4),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => ([(0,6),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => ([(0,3),(0,4),(0,6),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(0,7),(1,3),(1,4),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
Description
The size of the minimal feedback vertex set.
A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
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