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Your data matches 84 different statistics following compositions of up to 3 maps.
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Matching statistic: St000352
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(load all 6 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St001135
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000678
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 0 + 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001199
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
=> [1,0]
=> [2,1] => [1,1,0,0]
=> ? = 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> ? = 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,1,1}
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,1,1}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,1,1}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,1,1}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,1,1}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[(1,4),(2,6),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,5),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,4),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[(1,6),(2,4),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,4),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,4),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,4),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,4),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,4),(3,8),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,6),(3,8),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,7),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000755
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> []
=> ? ∊ {0,1,1}
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 1
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St001913
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> []
=> ? ∊ {0,1,1}
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
Description
The number of preimages of an integer partition in Bulgarian solitaire.
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
Partitions without preimages are called garden of eden partitions [1].
Matching statistic: St001195
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 67%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
=> [1,0]
=> []
=> []
=> ? = 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1}
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {0,1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {0,1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,6),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> []
=> => ? = 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> 10 => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> => ? ∊ {0,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> => ? ∊ {0,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 100 => 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 100 => 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 10 => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 10 => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> => ? ∊ {0,0,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> => ? ∊ {0,0,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> => ? ∊ {0,0,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 10 => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 110 => 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 110 => 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 10 => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> => ? ∊ {0,0,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> => ? ∊ {0,0,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> => ? ∊ {0,0,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 2
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 2
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,6),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 2
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0
[(1,4),(2,5),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,5),(2,4),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[(1,6),(2,4),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,7),(2,4),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,8),(2,4),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000618
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1,1}
[(1,3),(2,4)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
Description
The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000781
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 41%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 41%●distinct values known / distinct values provided: 33%
Values
[(1,2)]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1,1}
[(1,3),(2,4)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St001128The exponens consonantiae of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000701The protection number of a binary tree. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000660The number of rises of length at least 3 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001732The number of peaks visible from the left. St000007The number of saliances of the permutation. St000028The number of stack-sorts needed to sort a permutation. St000862The number of parts of the shifted shape of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001273The projective dimension of the first term in an injective coresolution of the regular module. St000451The length of the longest pattern of the form k 1 2. St000665The number of rafts of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000891The number of distinct diagonal sums of a permutation matrix. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000534The number of 2-rises of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000306The bounce count of a Dyck path. St000402Half the size of the symmetry class of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001394The genus of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000397The Strahler number of a rooted tree. St000675The number of centered multitunnels of a Dyck path. St000359The number of occurrences of the pattern 23-1. St000648The number of 2-excedences of a permutation. St000731The number of double exceedences of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000237The number of small exceedances. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001115The number of even descents of a permutation. St000054The first entry of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000366The number of double descents of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001096The size of the overlap set of a permutation.
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