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Matching statistic: St000048
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000048: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000048: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2]
=> 1
[[2,2]]
=> [2]
=> 1
[[1],[2]]
=> [1,1]
=> 2
[[1,3]]
=> [2]
=> 1
[[2,3]]
=> [2]
=> 1
[[3,3]]
=> [2]
=> 1
[[1],[3]]
=> [1,1]
=> 2
[[2],[3]]
=> [1,1]
=> 2
[[1,1,2]]
=> [3]
=> 1
[[1,2,2]]
=> [3]
=> 1
[[2,2,2]]
=> [3]
=> 1
[[1,1],[2]]
=> [2,1]
=> 3
[[1,2],[2]]
=> [2,1]
=> 3
[[1,4]]
=> [2]
=> 1
[[2,4]]
=> [2]
=> 1
[[3,4]]
=> [2]
=> 1
[[4,4]]
=> [2]
=> 1
[[1],[4]]
=> [1,1]
=> 2
[[2],[4]]
=> [1,1]
=> 2
[[3],[4]]
=> [1,1]
=> 2
[[1,1,3]]
=> [3]
=> 1
[[1,2,3]]
=> [3]
=> 1
[[1,3,3]]
=> [3]
=> 1
[[2,2,3]]
=> [3]
=> 1
[[2,3,3]]
=> [3]
=> 1
[[3,3,3]]
=> [3]
=> 1
[[1,1],[3]]
=> [2,1]
=> 3
[[1,2],[3]]
=> [2,1]
=> 3
[[1,3],[2]]
=> [2,1]
=> 3
[[1,3],[3]]
=> [2,1]
=> 3
[[2,2],[3]]
=> [2,1]
=> 3
[[2,3],[3]]
=> [2,1]
=> 3
[[1],[2],[3]]
=> [1,1,1]
=> 6
[[1,1,1,2]]
=> [4]
=> 1
[[1,1,2,2]]
=> [4]
=> 1
[[1,2,2,2]]
=> [4]
=> 1
[[2,2,2,2]]
=> [4]
=> 1
[[1,1,1],[2]]
=> [3,1]
=> 4
[[1,1,2],[2]]
=> [3,1]
=> 4
[[1,2,2],[2]]
=> [3,1]
=> 4
[[1,1],[2,2]]
=> [2,2]
=> 6
[[1,5]]
=> [2]
=> 1
[[2,5]]
=> [2]
=> 1
[[3,5]]
=> [2]
=> 1
[[4,5]]
=> [2]
=> 1
[[5,5]]
=> [2]
=> 1
[[1],[5]]
=> [1,1]
=> 2
[[2],[5]]
=> [1,1]
=> 2
[[3],[5]]
=> [1,1]
=> 2
[[4],[5]]
=> [1,1]
=> 2
Description
The multinomial of the parts of a partition.
Given an integer partition $\lambda = [\lambda_1,\ldots,\lambda_k]$, this is the multinomial
$$\binom{|\lambda|}{\lambda_1,\ldots,\lambda_k}.$$
For any integer composition $\mu$ that is a rearrangement of $\lambda$, this is the number of ordered set partitions whose list of block sizes is $\mu$.
Matching statistic: St000110
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[2,2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[1],[2]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[1,3]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[2,3]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[3,3]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[1],[3]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[2],[3]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[1,1,2]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[1,2,2]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[2,2,2]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[1,1],[2]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[1,2],[2]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[1,4]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[2,4]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[3,4]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[4,4]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[1],[4]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[2],[4]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[3],[4]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[1,1,3]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[1,2,3]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[1,3,3]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[2,2,3]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[2,3,3]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[3,3,3]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[1,1],[3]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[1,2],[3]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[1,3],[2]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[1,3],[3]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[2,2],[3]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[2,3],[3]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[1],[2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 6
[[1,1,1,2]]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[[1,1,2,2]]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[[1,2,2,2]]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[[2,2,2,2]]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[[1,1,1],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 4
[[1,1,2],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 4
[[1,2,2],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 4
[[1,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 6
[[1,5]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[2,5]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[3,5]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[4,5]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[5,5]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[1],[5]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[2],[5]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[3],[5]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[4],[5]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000085
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000085: Ordered trees ⟶ ℤResult quality: 70% ●values known / values provided: 95%●distinct values known / distinct values provided: 70%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000085: Ordered trees ⟶ ℤResult quality: 70% ●values known / values provided: 95%●distinct values known / distinct values provided: 70%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 3
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 6
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 4
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 4
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 6
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2
[[1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [[[[]]],[[[[]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [[[[]]],[[[[]]]]]
=> ? ∊ {1,1,1,1,1,1,1,35,35}
[[1,1,1,1,1,1,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,1,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,1,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[3,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,1],[2],[3]]
=> [7,6,1,2,3,4,5] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,2],[2],[3]]
=> [7,5,1,2,3,4,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,3],[2],[3]]
=> [6,5,1,2,3,4,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,2,2],[2],[3]]
=> [7,4,1,2,3,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,2,3],[2],[3]]
=> [6,4,1,2,3,5,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,3,3],[2],[3]]
=> [5,4,1,2,3,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,2,2],[2],[3]]
=> [7,3,1,2,4,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,2,3],[2],[3]]
=> [6,3,1,2,4,5,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,3,3],[2],[3]]
=> [5,3,1,2,4,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,3,3,3],[2],[3]]
=> [4,3,1,2,5,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,2,2],[2],[3]]
=> [7,2,1,3,4,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,2,3],[2],[3]]
=> [6,2,1,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,3,3],[2],[3]]
=> [5,2,1,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[],[],[[[[[]]]]]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
Description
The number of linear extensions of the tree.
We use Knuth's hook length formula for trees [pg.70, 1]. For an ordered tree $T$ on $n$ vertices, the number of linear extensions is
$$
\frac{n!}{\prod_{v\in T}|T_v|},
$$
where $T_v$ is the number of vertices of the subtree rooted at $v$.
Matching statistic: St000014
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000014: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 91%●distinct values known / distinct values provided: 59%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000014: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 91%●distinct values known / distinct values provided: 59%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[2,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[1,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[2,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[3,3]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[1],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[2],[3]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[1,1,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[1,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[2,2,2]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[1,1],[2]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[[1,2],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[[1,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[2,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[3,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[4,4]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[1],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[2],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[3],[4]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[1,1,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[1,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[2,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[2,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[3,3,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[[1,1],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[[1,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[[2,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[[2,3],[3]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 6
[[1,1,1,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 4
[[1,2,2],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 4
[[1,1],[2,2]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 6
[[1,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[2,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[3,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[4,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[5,5]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[[1],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[2],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[3],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[4],[5]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[[1,1,1,1,1,1,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1,1,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[2,2,2,2,2,2,2]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1,1,1],[2]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1,1,2],[2]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1,2,2],[2]]
=> [5,1,2,3,4,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,2,2,2],[2]]
=> [4,1,2,3,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,2,2,2,2],[2]]
=> [3,1,2,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,2,2,2,2,2],[2]]
=> [2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1,1],[2,2]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1,2],[2,2]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,2,2],[2,2]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,2,2,2],[2,2]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1],[2,2,2]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,7,7,7,7,7,7,21,21,21,21,35,35}
[[1,1,1,1,1,1,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,1,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,1,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,2,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,2,2,2,2,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,2,2,2,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,2,2,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,2,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,2,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[2,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[3,3,3,3,3,3,3]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,1,1],[3]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,1,2],[3]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
[[1,1,1,1,1,3],[2]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,35,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,105,105,105,105,105,105,105,105,105,105,105,105,105,105,105,140,140,140,140,140,140,210,210,210}
Description
The number of parking functions supported by a Dyck path.
One representation of a parking function is as a pair consisting of a Dyck path and a permutation $\pi$ such that if $[a_0, a_1, \dots, a_{n-1}]$ is the area sequence of the Dyck path then the permutation $\pi$ satisfies $pi_i < pi_{i+1}$ whenever $a_{i} < a_{i+1}$. This statistic counts the number of permutations $\pi$ which satisfy this condition.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 38%●distinct values known / distinct values provided: 26%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 38%●distinct values known / distinct values provided: 26%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,6} - 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,6} - 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,6} - 1
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,6} - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,6} - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,6} - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,6,6,6} - 1
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,1,3],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,1,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[2,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,4,4,6,6,6,6,6,12,12,12} - 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {5,5,10,10} - 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {5,5,10,10} - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {5,5,10,10} - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {5,5,10,10} - 1
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[3,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[4,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[2],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[2],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[3],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6} - 1
[[1,1,1],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,12,12,12,12,12,12,12,12,12,12,24} - 1
[[1,1,2],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,12,12,12,12,12,12,12,12,12,12,24} - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 24%●distinct values known / distinct values provided: 19%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 24%●distinct values known / distinct values provided: 19%
Values
[[1,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,3]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[3]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3} + 1
[[1,2],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3} + 1
[[1,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4,4]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[4]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,6} + 1
[[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,6} + 1
[[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,6} + 1
[[1,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,6} + 1
[[2,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,6} + 1
[[2,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,6} + 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,6} + 1
[[1,1,2],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,6} + 1
[[1,2,2],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,6} + 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,6} + 1
[[1,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[5,5]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[2],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[3],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[4],[5]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,1,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,3,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[4,4,4]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[1,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[1,4],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[1,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[1,4],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[1,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[2,2],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[2,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[2,4],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[2,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[3,3],[4]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[3,4],[4]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,3,3,3,3,3,3,3,6,6,6} + 1
[[1],[2],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1],[3],[4]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,1,1],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,1,3],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,1,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,2,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,3,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[2,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[2,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[2,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,12,12,12} + 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,5,5,5,10,10} + 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,5,5,5,10,10} + 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,5,5,5,10,10} + 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {5,5,5,5,10,10} + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {5,5,5,5,10,10} + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {5,5,5,5,10,10} + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001568
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 9%●distinct values known / distinct values provided: 4%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 9%●distinct values known / distinct values provided: 4%
Values
[[1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,2}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,2}
[[1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,2}
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,2,2}
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1],[3]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,2,2}
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,2,2}
[[1,1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,3,3}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,3,3}
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,3,3}
[[1,1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,1,3,3}
[[1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,3,3}
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,2,2,2}
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {1,2,2,2}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,2,2,2}
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[1,1],[3]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[1],[2],[3]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,3,3,6}
[[1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,1,4,4,4,6}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {1,1,1,1,2,2,2,2}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [9,9,9,9,3,3]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 1
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 1
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,5],[3]]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> [3,2]
=> 1
[[1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[2],[3],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 1
[[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 1
[[1,2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,2,4],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 1
[[1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 1
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 1
[[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
[[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[1,2,2,3],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,2}
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2}
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2}
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,2,2}
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,3,3}
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,3,3}
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3}
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,2,2,2}
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,2,2,2}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,2,2,2}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,3,3,3,3,3,3,6}
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,4,4,4,6}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,4,4,4,6}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,4,4,4,6}
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,4,4,4,6}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,4,4,4,6}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,4,4,4,6}
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,2,2,2,2}
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,2,2,2,2}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,2,2}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,2}
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,2,2,2,2}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,2,2,2,2}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,23),(1,24),(1,25),(1,26),(1,27),(1,28),(2,18),(2,19),(2,20),(2,21),(2,22),(2,28),(3,14),(3,15),(3,16),(3,17),(3,22),(3,27),(4,11),(4,12),(4,13),(4,17),(4,21),(4,26),(5,9),(5,10),(5,13),(5,16),(5,20),(5,25),(6,8),(6,10),(6,12),(6,15),(6,19),(6,24),(7,8),(7,9),(7,11),(7,14),(7,18),(7,23),(8,29),(8,32),(8,35),(8,41),(8,51),(9,29),(9,30),(9,33),(9,39),(9,49),(10,29),(10,31),(10,34),(10,40),(10,50),(11,30),(11,32),(11,36),(11,42),(11,52),(12,31),(12,32),(12,37),(12,43),(12,53),(13,30),(13,31),(13,38),(13,44),(13,54),(14,33),(14,35),(14,36),(14,45),(14,55),(15,34),(15,35),(15,37),(15,46),(15,56),(16,33),(16,34),(16,38),(16,47),(16,57),(17,36),(17,37),(17,38),(17,48),(17,58),(18,39),(18,41),(18,42),(18,45),(18,59),(19,40),(19,41),(19,43),(19,46),(19,60),(20,39),(20,40),(20,44),(20,47),(20,61),(21,42),(21,43),(21,44),(21,48),(21,62),(22,45),(22,46),(22,47),(22,48),(22,63),(23,49),(23,51),(23,52),(23,55),(23,59),(24,50),(24,51),(24,53),(24,56),(24,60),(25,49),(25,50),(25,54),(25,57),(25,61),(26,52),(26,53),(26,54),(26,58),(26,62),(27,55),(27,56),(27,57),(27,58),(27,63),(28,59),(28,60),(28,61),(28,62),(28,63),(29,67),(29,68),(29,78),(29,88),(30,64),(30,68),(30,75),(30,85),(31,65),(31,68),(31,76),(31,86),(32,66),(32,68),(32,77),(32,87),(33,64),(33,67),(33,69),(33,79),(34,65),(34,67),(34,70),(34,80),(35,66),(35,67),(35,71),(35,81),(36,64),(36,66),(36,72),(36,82),(37,65),(37,66),(37,73),(37,83),(38,64),(38,65),(38,74),(38,84),(39,69),(39,75),(39,78),(39,89),(40,70),(40,76),(40,78),(40,90),(41,71),(41,77),(41,78),(41,91),(42,72),(42,75),(42,77),(42,92),(43,73),(43,76),(43,77),(43,93),(44,74),(44,75),(44,76),(44,94),(45,69),(45,71),(45,72),(45,95),(46,70),(46,71),(46,73),(46,96),(47,69),(47,70),(47,74),(47,97),(48,72),(48,73),(48,74),(48,98),(49,79),(49,85),(49,88),(49,89),(50,80),(50,86),(50,88),(50,90),(51,81),(51,87),(51,88),(51,91),(52,82),(52,85),(52,87),(52,92),(53,83),(53,86),(53,87),(53,93),(54,84),(54,85),(54,86),(54,94),(55,79),(55,81),(55,82),(55,95),(56,80),(56,81),(56,83),(56,96),(57,79),(57,80),(57,84),(57,97),(58,82),(58,83),(58,84),(58,98),(59,89),(59,91),(59,92),(59,95),(60,90),(60,91),(60,93),(60,96),(61,89),(61,90),(61,94),(61,97),(62,92),(62,93),(62,94),(62,98),(63,95),(63,96),(63,97),(63,98),(64,109),(64,114),(64,119),(65,110),(65,115),(65,119),(66,111),(66,116),(66,119),(67,112),(67,117),(67,119),(68,113),(68,118),(68,119),(69,99),(69,114),(69,117),(70,100),(70,115),(70,117),(71,101),(71,116),(71,117),(72,102),(72,114),(72,116),(73,103),(73,115),(73,116),(74,104),(74,114),(74,115),(75,105),(75,114),(75,118),(76,106),(76,115),(76,118),(77,107),(77,116),(77,118),(78,108),(78,117),(78,118),(79,99),(79,109),(79,112),(80,100),(80,110),(80,112),(81,101),(81,111),(81,112),(82,102),(82,109),(82,111),(83,103),(83,110),(83,111),(84,104),(84,109),(84,110),(85,105),(85,109),(85,113),(86,106),(86,110),(86,113),(87,107),(87,111),(87,113),(88,108),(88,112),(88,113),(89,99),(89,105),(89,108),(90,100),(90,106),(90,108),(91,101),(91,107),(91,108),(92,102),(92,105),(92,107),(93,103),(93,106),(93,107),(94,104),(94,105),(94,106),(95,99),(95,101),(95,102),(96,100),(96,101),(96,103),(97,99),(97,100),(97,104),(98,102),(98,103),(98,104),(99,120),(99,123),(100,121),(100,123),(101,122),(101,123),(102,120),(102,122),(103,121),(103,122),(104,120),(104,121),(105,120),(105,124),(106,121),(106,124),(107,122),(107,124),(108,123),(108,124),(109,120),(109,125),(110,121),(110,125),(111,122),(111,125),(112,123),(112,125),(113,124),(113,125),(114,120),(114,126),(115,121),(115,126),(116,122),(116,126),(117,123),(117,126),(118,124),(118,126),(119,125),(119,126),(120,127),(121,127),(122,127),(123,127),(124,127),(125,127),(126,127)],128)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,23),(1,24),(1,25),(1,26),(1,27),(1,28),(2,18),(2,19),(2,20),(2,21),(2,22),(2,28),(3,14),(3,15),(3,16),(3,17),(3,22),(3,27),(4,11),(4,12),(4,13),(4,17),(4,21),(4,26),(5,9),(5,10),(5,13),(5,16),(5,20),(5,25),(6,8),(6,10),(6,12),(6,15),(6,19),(6,24),(7,8),(7,9),(7,11),(7,14),(7,18),(7,23),(8,29),(8,32),(8,35),(8,41),(8,51),(9,29),(9,30),(9,33),(9,39),(9,49),(10,29),(10,31),(10,34),(10,40),(10,50),(11,30),(11,32),(11,36),(11,42),(11,52),(12,31),(12,32),(12,37),(12,43),(12,53),(13,30),(13,31),(13,38),(13,44),(13,54),(14,33),(14,35),(14,36),(14,45),(14,55),(15,34),(15,35),(15,37),(15,46),(15,56),(16,33),(16,34),(16,38),(16,47),(16,57),(17,36),(17,37),(17,38),(17,48),(17,58),(18,39),(18,41),(18,42),(18,45),(18,59),(19,40),(19,41),(19,43),(19,46),(19,60),(20,39),(20,40),(20,44),(20,47),(20,61),(21,42),(21,43),(21,44),(21,48),(21,62),(22,45),(22,46),(22,47),(22,48),(22,63),(23,49),(23,51),(23,52),(23,55),(23,59),(24,50),(24,51),(24,53),(24,56),(24,60),(25,49),(25,50),(25,54),(25,57),(25,61),(26,52),(26,53),(26,54),(26,58),(26,62),(27,55),(27,56),(27,57),(27,58),(27,63),(28,59),(28,60),(28,61),(28,62),(28,63),(29,67),(29,68),(29,78),(29,88),(30,64),(30,68),(30,75),(30,85),(31,65),(31,68),(31,76),(31,86),(32,66),(32,68),(32,77),(32,87),(33,64),(33,67),(33,69),(33,79),(34,65),(34,67),(34,70),(34,80),(35,66),(35,67),(35,71),(35,81),(36,64),(36,66),(36,72),(36,82),(37,65),(37,66),(37,73),(37,83),(38,64),(38,65),(38,74),(38,84),(39,69),(39,75),(39,78),(39,89),(40,70),(40,76),(40,78),(40,90),(41,71),(41,77),(41,78),(41,91),(42,72),(42,75),(42,77),(42,92),(43,73),(43,76),(43,77),(43,93),(44,74),(44,75),(44,76),(44,94),(45,69),(45,71),(45,72),(45,95),(46,70),(46,71),(46,73),(46,96),(47,69),(47,70),(47,74),(47,97),(48,72),(48,73),(48,74),(48,98),(49,79),(49,85),(49,88),(49,89),(50,80),(50,86),(50,88),(50,90),(51,81),(51,87),(51,88),(51,91),(52,82),(52,85),(52,87),(52,92),(53,83),(53,86),(53,87),(53,93),(54,84),(54,85),(54,86),(54,94),(55,79),(55,81),(55,82),(55,95),(56,80),(56,81),(56,83),(56,96),(57,79),(57,80),(57,84),(57,97),(58,82),(58,83),(58,84),(58,98),(59,89),(59,91),(59,92),(59,95),(60,90),(60,91),(60,93),(60,96),(61,89),(61,90),(61,94),(61,97),(62,92),(62,93),(62,94),(62,98),(63,95),(63,96),(63,97),(63,98),(64,109),(64,114),(64,119),(65,110),(65,115),(65,119),(66,111),(66,116),(66,119),(67,112),(67,117),(67,119),(68,113),(68,118),(68,119),(69,99),(69,114),(69,117),(70,100),(70,115),(70,117),(71,101),(71,116),(71,117),(72,102),(72,114),(72,116),(73,103),(73,115),(73,116),(74,104),(74,114),(74,115),(75,105),(75,114),(75,118),(76,106),(76,115),(76,118),(77,107),(77,116),(77,118),(78,108),(78,117),(78,118),(79,99),(79,109),(79,112),(80,100),(80,110),(80,112),(81,101),(81,111),(81,112),(82,102),(82,109),(82,111),(83,103),(83,110),(83,111),(84,104),(84,109),(84,110),(85,105),(85,109),(85,113),(86,106),(86,110),(86,113),(87,107),(87,111),(87,113),(88,108),(88,112),(88,113),(89,99),(89,105),(89,108),(90,100),(90,106),(90,108),(91,101),(91,107),(91,108),(92,102),(92,105),(92,107),(93,103),(93,106),(93,107),(94,104),(94,105),(94,106),(95,99),(95,101),(95,102),(96,100),(96,101),(96,103),(97,99),(97,100),(97,104),(98,102),(98,103),(98,104),(99,120),(99,123),(100,121),(100,123),(101,122),(101,123),(102,120),(102,122),(103,121),(103,122),(104,120),(104,121),(105,120),(105,124),(106,121),(106,124),(107,122),(107,124),(108,123),(108,124),(109,120),(109,125),(110,121),(110,125),(111,122),(111,125),(112,123),(112,125),(113,124),(113,125),(114,120),(114,126),(115,121),(115,126),(116,122),(116,126),(117,123),(117,126),(118,124),(118,126),(119,125),(119,126),(120,127),(121,127),(122,127),(123,127),(124,127),(125,127),(126,127)],128)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1],[3,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2],[2,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,3],[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,5],[3]]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1],[2],[5]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2],[3],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2,4],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,3,4],[3]]
=> ([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2,4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[2,3],[4,4]]
=> ([(0,2),(1,8),(2,5),(2,6),(2,7),(3,17),(4,16),(5,12),(5,13),(6,12),(6,14),(7,13),(7,14),(8,10),(8,11),(9,18),(10,18),(11,18),(12,1),(13,4),(13,15),(14,3),(14,15),(15,16),(15,17),(16,9),(16,10),(17,9),(17,11)],19)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,3],[4,4]]
=> ([(0,9),(1,10),(1,18),(2,10),(2,17),(3,17),(3,18),(5,14),(6,15),(7,12),(7,13),(8,7),(9,1),(9,2),(9,3),(10,8),(11,14),(11,15),(12,19),(13,19),(14,12),(14,16),(15,13),(15,16),(16,19),(17,5),(17,11),(18,6),(18,11),(19,4)],20)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,2,2,3],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,1],[3,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[1,1,2],[2,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,2}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,2}
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2}
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2}
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,2,2}
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2}
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,2,2,2}
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,2,2,2}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,2,2,2}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[[1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,4],[2,3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[[1,1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2],[4]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001118
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,2}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,2}
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2}
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2}
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,2,2}
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2}
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,3,3}
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,2,2,2}
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,2,2,2}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,2,2,2}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,6}
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,4,4,4,6}
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6}
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[[1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,4],[2,3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[[1,1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2],[4]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000467The hyper-Wiener index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001545The second Elser number of a connected graph. St001645The pebbling number of a connected graph. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000103The sum of the entries of a semistandard tableau. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
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