Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000460
St000460: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> 2 = 1 + 1
[1,1]
=> 2 = 1 + 1
[3]
=> 3 = 2 + 1
[2,1]
=> 3 = 2 + 1
[1,1,1]
=> 3 = 2 + 1
[4]
=> 4 = 3 + 1
[3,1]
=> 4 = 3 + 1
[2,2]
=> 1 = 0 + 1
[2,1,1]
=> 4 = 3 + 1
[1,1,1,1]
=> 4 = 3 + 1
[5]
=> 5 = 4 + 1
[4,1]
=> 5 = 4 + 1
[3,2]
=> 1 = 0 + 1
[3,1,1]
=> 5 = 4 + 1
[2,2,1]
=> 1 = 0 + 1
[2,1,1,1]
=> 5 = 4 + 1
[1,1,1,1,1]
=> 5 = 4 + 1
[6]
=> 6 = 5 + 1
[5,1]
=> 6 = 5 + 1
[4,2]
=> 1 = 0 + 1
[4,1,1]
=> 6 = 5 + 1
[3,3]
=> 2 = 1 + 1
[3,2,1]
=> 1 = 0 + 1
[3,1,1,1]
=> 6 = 5 + 1
[2,2,2]
=> 2 = 1 + 1
[2,2,1,1]
=> 1 = 0 + 1
[2,1,1,1,1]
=> 6 = 5 + 1
[1,1,1,1,1,1]
=> 6 = 5 + 1
[7]
=> 7 = 6 + 1
[6,1]
=> 7 = 6 + 1
[5,2]
=> 1 = 0 + 1
[5,1,1]
=> 7 = 6 + 1
[4,3]
=> 2 = 1 + 1
[4,2,1]
=> 1 = 0 + 1
[4,1,1,1]
=> 7 = 6 + 1
[3,3,1]
=> 2 = 1 + 1
[3,2,2]
=> 2 = 1 + 1
[3,2,1,1]
=> 1 = 0 + 1
[3,1,1,1,1]
=> 7 = 6 + 1
[2,2,2,1]
=> 2 = 1 + 1
[2,2,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> 7 = 6 + 1
[1,1,1,1,1,1,1]
=> 7 = 6 + 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000989
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000989: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 3
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 4
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 4
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 5
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5,6] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,4,5,6] => 5
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,3,4,5,6] => 5
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,2,3,4,5,6] => 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 5
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 6
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,4,6,5] => 0
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => 6
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,3,5,6,4] => 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => 6
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [1,2,4,5,6,3] => 0
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => 6
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => 6
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 6
Description
The number of final rises of a permutation. For a permutation $\pi$ of length $n$, this is the maximal $k$ such that $$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$ Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St000461
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000461: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 2 = 1 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [1,2] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3 = 2 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 3 = 2 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 4 = 3 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 4 = 3 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 5 = 4 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 5 = 4 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 5 = 4 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 5 = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6 = 5 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5,6] => 6 = 5 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 1 = 0 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,4,5,6] => 6 = 5 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,3,4,5,6] => 6 = 5 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,2,3,4,5,6] => 6 = 5 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6 = 5 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7 = 6 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => 7 = 6 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,4,6,5] => 1 = 0 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => 7 = 6 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 2 = 1 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,3,5,6,4] => 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => 7 = 6 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => 2 = 1 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [1,2,4,5,6,3] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => 7 = 6 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => 2 = 1 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => 7 = 6 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 7 = 6 + 1
Description
The rix statistic of a permutation. This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then $rix(w) := 0$ if $i = 1 < k$, $rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and $rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00140: Dyck paths logarithmic height to pruning numberBinary trees
Mp00013: Binary trees to posetPosets
St001879: Posets ⟶ ℤResult quality: 58% values known / values provided: 58%distinct values known / distinct values provided: 71%
Values
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1}
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1}
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,0}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,0,0,1,1}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,1,1}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,1,1}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,1,1}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {0,0,0,0,1,1,1,1}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {0,0,0,0,1,1,1,1}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {0,0,0,0,1,1,1,1}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,0,0,0,1,1,1,1}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00140: Dyck paths logarithmic height to pruning numberBinary trees
Mp00013: Binary trees to posetPosets
St001880: Posets ⟶ ℤResult quality: 58% values known / values provided: 58%distinct values known / distinct values provided: 71%
Values
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1} + 1
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1} + 1
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,0} + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0} + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,0,0,1,1} + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,1,1} + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,1,1} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,1,1} + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1} + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {0,0,0,0,1,1,1,1} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {0,0,0,0,1,1,1,1} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1} + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1} + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {0,0,0,0,1,1,1,1} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1} + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,0,0,0,1,1,1,1} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001772: Signed permutations ⟶ ℤResult quality: 44% values known / values provided: 44%distinct values known / distinct values provided: 86%
Values
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 4 = 3 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 4 = 3 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 4 = 3 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? ∊ {0,0,4} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => ? ∊ {0,0,4} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 5 = 4 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 5 = 4 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 5 = 4 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,3,2] => 5 = 4 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? ∊ {0,0,4} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? ∊ {0,0,0,1,1,5,5,5} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [4,5,3,2,1,6] => ? ∊ {0,0,0,1,1,5,5,5} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => ? ∊ {0,0,0,1,1,5,5,5} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {0,0,0,1,1,5,5,5} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {0,0,0,1,1,5,5,5} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 6 = 5 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,5,2] => 6 = 5 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {0,0,0,1,1,5,5,5} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,4,2] => 6 = 5 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [1,5,6,4,3,2] => ? ∊ {0,0,0,1,1,5,5,5} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? ∊ {0,0,0,1,1,5,5,5} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [5,6,4,3,2,1,7] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [4,3,5,2,1,6] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [3,5,4,2,1,6] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => 7 = 6 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 7 = 6 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [1,5,4,6,3,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => 7 = 6 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [1,4,6,5,3,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [1,6,7,5,4,3,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6} + 1
Description
The number of occurrences of the signed pattern 12 in a signed permutation. This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < \pi(j)$.
Matching statistic: St001384
Mp00202: Integer partitions first row removalInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001384: Integer partitions ⟶ ℤResult quality: 29% values known / values provided: 37%distinct values known / distinct values provided: 29%
Values
[2]
=> []
=> []
=> ?
=> ? ∊ {1,1}
[1,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1}
[3]
=> []
=> []
=> ?
=> ? ∊ {2,2,2}
[2,1]
=> [1]
=> [1]
=> []
=> ? ∊ {2,2,2}
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {2,2,2}
[4]
=> []
=> []
=> ?
=> ? ∊ {3,3,3,3}
[3,1]
=> [1]
=> [1]
=> []
=> ? ∊ {3,3,3,3}
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 0
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {3,3,3,3}
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {3,3,3,3}
[5]
=> []
=> []
=> ?
=> ? ∊ {4,4,4,4,4}
[4,1]
=> [1]
=> [1]
=> []
=> ? ∊ {4,4,4,4,4}
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 0
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {4,4,4,4,4}
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {4,4,4,4,4}
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {4,4,4,4,4}
[6]
=> []
=> []
=> ?
=> ? ∊ {5,5,5,5,5,5}
[5,1]
=> [1]
=> [1]
=> []
=> ? ∊ {5,5,5,5,5,5}
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 0
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {5,5,5,5,5,5}
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {5,5,5,5,5,5}
[2,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {5,5,5,5,5,5}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {5,5,5,5,5,5}
[7]
=> []
=> []
=> ?
=> ? ∊ {6,6,6,6,6,6,6}
[6,1]
=> [1]
=> [1]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 0
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St001767
Mp00202: Integer partitions first row removalInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001767: Integer partitions ⟶ ℤResult quality: 29% values known / values provided: 37%distinct values known / distinct values provided: 29%
Values
[2]
=> []
=> []
=> ?
=> ? ∊ {1,1}
[1,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1}
[3]
=> []
=> []
=> ?
=> ? ∊ {2,2,2}
[2,1]
=> [1]
=> [1]
=> []
=> ? ∊ {2,2,2}
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {2,2,2}
[4]
=> []
=> []
=> ?
=> ? ∊ {3,3,3,3}
[3,1]
=> [1]
=> [1]
=> []
=> ? ∊ {3,3,3,3}
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 0
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {3,3,3,3}
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {3,3,3,3}
[5]
=> []
=> []
=> ?
=> ? ∊ {4,4,4,4,4}
[4,1]
=> [1]
=> [1]
=> []
=> ? ∊ {4,4,4,4,4}
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 0
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {4,4,4,4,4}
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {4,4,4,4,4}
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {4,4,4,4,4}
[6]
=> []
=> []
=> ?
=> ? ∊ {5,5,5,5,5,5}
[5,1]
=> [1]
=> [1]
=> []
=> ? ∊ {5,5,5,5,5,5}
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 0
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {5,5,5,5,5,5}
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {5,5,5,5,5,5}
[2,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {5,5,5,5,5,5}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {5,5,5,5,5,5}
[7]
=> []
=> []
=> ?
=> ? ∊ {6,6,6,6,6,6,6}
[6,1]
=> [1]
=> [1]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 0
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> ? ∊ {6,6,6,6,6,6,6}
Description
The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
St001583: Permutations ⟶ ℤResult quality: 28% values known / values provided: 28%distinct values known / distinct values provided: 71%
Values
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 2 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3 = 2 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ? ∊ {0,3} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 3 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 3 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ? ∊ {0,3} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ? ∊ {0,0,4,4} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? ∊ {0,0,4,4} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5 = 4 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5 = 4 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5 = 4 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {0,0,4,4} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {0,0,4,4} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6 = 5 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,0,1,1,5,5,5,5,5} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,5,3,4,2,1,7] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,1,1,1,1,6,6,6,6,6,6,6} + 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Mp00202: Integer partitions first row removalInteger partitions
Mp00313: Integer partitions Glaisher-Franklin inverseInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000512: Integer partitions ⟶ ℤResult quality: 28% values known / values provided: 28%distinct values known / distinct values provided: 29%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,1}
[1,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1}
[3]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2}
[2,1]
=> [1]
=> [1]
=> []
=> ? ∊ {2,2,2}
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {2,2,2}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,3,3,3,3}
[3,1]
=> [1]
=> [1]
=> []
=> ? ∊ {0,3,3,3,3}
[2,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {0,3,3,3,3}
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,3,3,3,3}
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {0,3,3,3,3}
[5]
=> []
=> ?
=> ?
=> ? ∊ {4,4,4,4,4}
[4,1]
=> [1]
=> [1]
=> []
=> ? ∊ {4,4,4,4,4}
[3,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {4,4,4,4,4}
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {4,4,4,4,4}
[2,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {4,4,4,4,4}
[1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [2]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,5,5,5,5,5,5}
[5,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,5,5,5,5,5,5}
[4,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {1,5,5,5,5,5,5}
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,5,5,5,5,5,5}
[3,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,5,5,5,5,5,5}
[3,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,5,5,5,5,5,5}
[2,2,2]
=> [2,2]
=> [4]
=> []
=> ? ∊ {1,5,5,5,5,5,5}
[2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [2]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [2,2,1]
=> [2,1]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[6,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[5,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[4,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[4,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[3,3,1]
=> [3,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[3,2,2]
=> [2,2]
=> [4]
=> []
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[3,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> [2]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [4,1]
=> [1]
=> ? ∊ {1,1,6,6,6,6,6,6,6}
[2,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [2,2,2]
=> [2,2]
=> 0
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000941The number of characters of the symmetric group whose value on the partition is even. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000719The number of alignments in a perfect matching. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001626The number of maximal proper sublattices of a lattice. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001926Sparre Andersen's position of the maximum of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.