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Your data matches 200 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
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Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00065: Permutations āpermutation posetā¶ Posets
St001880: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00065: Permutations āpermutation posetā¶ Posets
St001880: Posets ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 4
[5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,6,3,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 6
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 6
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 6
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 4
[6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 6
[5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[6,5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 5
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,4,5,2,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 6
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 6
[6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> 5
[6,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,2,5,6,3,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 6
[6,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 6
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 5
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,2,5,3,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
[6,5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
[6,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
[6,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[6,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[6,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[6,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[6,5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000390
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00134: Standard tableaux ādescent wordā¶ Binary words
Mp00234: Binary words āvalleys-to-peaksā¶ Binary words
St000390: Binary words ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Mp00134: Standard tableaux ādescent wordā¶ Binary words
Mp00234: Binary words āvalleys-to-peaksā¶ Binary words
St000390: Binary words ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Values
[2,1]
=> [[1,3],[2]]
=> 10 => 11 => 1 = 3 - 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1101 => 2 = 4 - 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 11001 => 2 = 4 - 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 111000 => 111001 => 2 = 4 - 2
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 1101000 => 1110001 => 2 = 4 - 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> 11110000 => 11110001 => 2 = 4 - 2
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> 11001000 => 11010001 => 3 = 5 - 2
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> 10101000 => 11010001 => 3 = 5 - 2
[5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> 111010000 => 111100001 => 2 = 4 - 2
[4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> 101001000 => 110010001 => 3 = 5 - 2
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 4 - 2
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> 1110010000 => ? => ? = 6 - 2
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> 1101010000 => ? => ? = 6 - 2
[6,2,1,1,1,1]
=> [[1,6,9,10,11,12],[2,8],[3],[4],[5],[7]]
=> ? => ? => ? = 4 - 2
[5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> 11100010000 => ? => ? = 5 - 2
[5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> 11010010000 => ? => ? = 4 - 2
[5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> 10101010000 => ? => ? = 5 - 2
[6,3,1,1,1,1]
=> [[1,6,7,11,12,13],[2,9,10],[3],[4],[5],[8]]
=> ? => ? => ? = 6 - 2
[6,2,2,1,1,1]
=> [[1,5,10,11,12,13],[2,7],[3,9],[4],[6],[8]]
=> ? => ? => ? = 6 - 2
[5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> 110100010000 => ? => ? = 5 - 2
[5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? => ? = 5 - 2
[6,4,1,1,1,1]
=> [[1,6,7,8,13,14],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? => ? = 6 - 2
[6,3,2,1,1,1]
=> [[1,5,8,12,13,14],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? => ? = 4 - 2
[6,2,2,2,1,1]
=> [[1,4,11,12,13,14],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? => ? = 6 - 2
[5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> 1100100010000 => ? => ? = 6 - 2
[5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> 1010100010000 => ? => ? = 6 - 2
[5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> 1010010010000 => ? => ? = 6 - 2
[6,5,1,1,1,1]
=> [[1,6,7,8,9,15],[2,11,12,13,14],[3],[4],[5],[10]]
=> ? => ? => ? = 5 - 2
[6,4,2,1,1,1]
=> [[1,5,8,9,14,15],[2,7,12,13],[3,11],[4],[6],[10]]
=> ? => ? => ? = 6 - 2
[6,3,2,2,1,1]
=> [[1,4,9,13,14,15],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? => ? => ? = 6 - 2
[6,2,2,2,2,1]
=> [[1,3,12,13,14,15],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 5 - 2
[5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> 10100100010000 => ? => ? = 6 - 2
[6,5,2,1,1,1]
=> [[1,5,8,9,10,16],[2,7,13,14,15],[3,12],[4],[6],[11]]
=> ? => ? => ? = 5 - 2
[6,4,3,1,1,1]
=> [[1,5,6,10,15,16],[2,8,9,14],[3,12,13],[4],[7],[11]]
=> ? => ? => ? = 6 - 2
[6,3,3,2,1,1]
=> [[1,4,7,14,15,16],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? => ? => ? = 6 - 2
[6,3,2,2,2,1]
=> [[1,3,10,14,15,16],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? => ? => ? = 5 - 2
[6,5,3,1,1,1]
=> [[1,5,6,10,11,17],[2,8,9,15,16],[3,13,14],[4],[7],[12]]
=> ? => ? => ? = 7 - 2
[6,5,2,2,1,1]
=> [[1,4,9,10,11,17],[2,6,14,15,16],[3,8],[5,13],[7],[12]]
=> ? => ? => ? = 7 - 2
[6,4,3,2,1,1]
=> [[1,4,7,11,16,17],[2,6,10,15],[3,9,14],[5,13],[8],[12]]
=> ? => ? => ? = 4 - 2
[6,4,2,2,2,1]
=> [[1,3,10,11,16,17],[2,5,14,15],[4,7],[6,9],[8,13],[12]]
=> ? => ? => ? = 7 - 2
[6,3,3,2,2,1]
=> [[1,3,8,15,16,17],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? => ? => ? = 7 - 2
[6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? => ? => ? = 7 - 2
[6,4,4,3,2,1]
=> [[1,3,6,10,19,20],[2,5,9,14],[4,8,13,18],[7,12,17],[11,16],[15]]
=> ? => ? => ? = 7 - 2
[6,5,3,3,2,1]
=> [[1,3,6,13,14,20],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
=> ? => ? => ? = 7 - 2
[6,4,3,3,2,1]
=> [[1,3,6,13,18,19],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
=> ? => ? => ? = 6 - 2
[6,3,3,3,2,1]
=> [[1,3,6,16,17,18],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? => ? => ? = 6 - 2
[6,5,4,2,2,1]
=> [[1,3,8,9,14,20],[2,5,12,13,19],[4,7,17,18],[6,11],[10,16],[15]]
=> ? => ? => ? = 7 - 2
[6,5,3,2,2,1]
=> [[1,3,8,12,13,19],[2,5,11,17,18],[4,7,16],[6,10],[9,15],[14]]
=> ? => ? => ? = 6 - 2
[6,4,3,2,2,1]
=> [[1,3,8,12,17,18],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
=> ? => ? => ? = 5 - 2
[6,5,2,2,2,1]
=> [[1,3,10,11,12,18],[2,5,15,16,17],[4,7],[6,9],[8,14],[13]]
=> ? => ? => ? = 6 - 2
[6,5,4,3,1,1]
=> [[1,4,5,9,14,20],[2,7,8,13,19],[3,11,12,18],[6,16,17],[10],[15]]
=> ? => ? => ? = 7 - 2
[6,4,4,3,1,1]
=> [[1,4,5,9,18,19],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
=> ? => ? => ? = 7 - 2
[6,5,4,2,1,1]
=> [[1,4,7,8,13,19],[2,6,11,12,18],[3,10,16,17],[5,15],[9],[14]]
=> ? => ? => ? = 6 - 2
[6,5,3,2,1,1]
=> [[1,4,7,11,12,18],[2,6,10,16,17],[3,9,15],[5,14],[8],[13]]
=> ? => ? => ? = 5 - 2
[6,5,4,1,1,1]
=> [[1,5,6,7,12,18],[2,9,10,11,17],[3,14,15,16],[4],[8],[13]]
=> ? => ? => ? = 6 - 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000473
Mp00179: Integer partitions āto skew partitionā¶ Skew partitions
Mp00185: Skew partitions ācell posetā¶ Posets
Mp00110: Posets āGreene-Kleitman invariantā¶ Integer partitions
St000473: Integer partitions ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Mp00185: Skew partitions ācell posetā¶ Posets
Mp00110: Posets āGreene-Kleitman invariantā¶ Integer partitions
St000473: Integer partitions ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Values
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 1 = 3 - 2
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 2 = 4 - 2
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [3,2,1]
=> 2 = 4 - 2
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> [4,3]
=> 2 = 4 - 2
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> [4,3,1]
=> 2 = 4 - 2
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
=> [5,4]
=> 2 = 4 - 2
[4,3,1,1]
=> [[4,3,1,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> [4,3,2]
=> 3 = 5 - 2
[4,2,2,1]
=> [[4,2,2,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> [4,3,2]
=> 3 = 5 - 2
[5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(7,9),(8,4),(8,9)],10)
=> [5,4,1]
=> 2 = 4 - 2
[4,3,2,1]
=> [[4,3,2,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> [4,3,2,1]
=> 3 = 5 - 2
[6,1,1,1,1,1]
=> [[6,1,1,1,1,1],[]]
=> ?
=> ?
=> ? = 4 - 2
[5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ?
=> ? = 6 - 2
[5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ?
=> ? = 6 - 2
[6,2,1,1,1,1]
=> [[6,2,1,1,1,1],[]]
=> ?
=> ?
=> ? = 4 - 2
[5,4,1,1,1]
=> [[5,4,1,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ?
=> ? = 5 - 2
[5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11)],12)
=> ?
=> ? = 4 - 2
[5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ?
=> ? = 5 - 2
[6,3,1,1,1,1]
=> [[6,3,1,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,2,2,1,1,1]
=> [[6,2,2,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ?
=> ? = 5 - 2
[5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ?
=> ? = 5 - 2
[6,4,1,1,1,1]
=> [[6,4,1,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,2,1,1,1]
=> [[6,3,2,1,1,1],[]]
=> ?
=> ?
=> ? = 4 - 2
[6,2,2,2,1,1]
=> [[6,2,2,2,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ?
=> ? = 6 - 2
[5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(13,11)],14)
=> ?
=> ? = 6 - 2
[5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ?
=> ? = 6 - 2
[6,5,1,1,1,1]
=> [[6,5,1,1,1,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,4,2,1,1,1]
=> [[6,4,2,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,2,2,1,1]
=> [[6,3,2,2,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,2,2,2,2,1]
=> [[6,2,2,2,2,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(12,14),(13,11),(13,14)],15)
=> ?
=> ? = 6 - 2
[6,5,2,1,1,1]
=> [[6,5,2,1,1,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,4,3,1,1,1]
=> [[6,4,3,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,3,2,1,1]
=> [[6,3,3,2,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,2,2,2,1]
=> [[6,3,2,2,2,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,5,3,1,1,1]
=> [[6,5,3,1,1,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,2,2,1,1]
=> [[6,5,2,2,1,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,4,3,2,1,1]
=> [[6,4,3,2,1,1],[]]
=> ?
=> ?
=> ? = 4 - 2
[6,4,2,2,2,1]
=> [[6,4,2,2,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,3,3,2,2,1]
=> [[6,3,3,2,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,4,3,2,1]
=> [[6,5,4,3,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,4,4,3,2,1]
=> [[6,4,4,3,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,3,3,2,1]
=> [[6,5,3,3,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,4,3,3,2,1]
=> [[6,4,3,3,2,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,3,3,2,1]
=> [[6,3,3,3,2,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,5,4,2,2,1]
=> [[6,5,4,2,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,3,2,2,1]
=> [[6,5,3,2,2,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,4,3,2,2,1]
=> [[6,4,3,2,2,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,5,2,2,2,1]
=> [[6,5,2,2,2,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,5,4,3,1,1]
=> [[6,5,4,3,1,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,4,4,3,1,1]
=> [[6,4,4,3,1,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,4,2,1,1]
=> [[6,5,4,2,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,5,3,2,1,1]
=> [[6,5,3,2,1,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,5,4,1,1,1]
=> [[6,5,4,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000630
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00134: Standard tableaux ādescent wordā¶ Binary words
Mp00280: Binary words āpath rowmotionā¶ Binary words
St000630: Binary words ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Mp00134: Standard tableaux ādescent wordā¶ Binary words
Mp00280: Binary words āpath rowmotionā¶ Binary words
St000630: Binary words ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Values
[2,1]
=> [[1,3],[2]]
=> 10 => 11 => 1 = 3 - 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0111 => 2 = 4 - 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 11001 => 2 = 4 - 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 111000 => 001111 => 2 = 4 - 2
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 1101000 => 1110001 => 2 = 4 - 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> 11110000 => 00011111 => 2 = 4 - 2
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> 11001000 => 01110001 => 3 = 5 - 2
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> 10101000 => 11010001 => 3 = 5 - 2
[5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> 111010000 => 111100001 => 2 = 4 - 2
[4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> 101001000 => 110010001 => 3 = 5 - 2
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 4 - 2
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> 1110010000 => ? => ? = 6 - 2
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> 1101010000 => ? => ? = 6 - 2
[6,2,1,1,1,1]
=> [[1,6,9,10,11,12],[2,8],[3],[4],[5],[7]]
=> ? => ? => ? = 4 - 2
[5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> 11100010000 => ? => ? = 5 - 2
[5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> 11010010000 => ? => ? = 4 - 2
[5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> 10101010000 => ? => ? = 5 - 2
[6,3,1,1,1,1]
=> [[1,6,7,11,12,13],[2,9,10],[3],[4],[5],[8]]
=> ? => ? => ? = 6 - 2
[6,2,2,1,1,1]
=> [[1,5,10,11,12,13],[2,7],[3,9],[4],[6],[8]]
=> ? => ? => ? = 6 - 2
[5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> 110100010000 => ? => ? = 5 - 2
[5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? => ? = 5 - 2
[6,4,1,1,1,1]
=> [[1,6,7,8,13,14],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? => ? = 6 - 2
[6,3,2,1,1,1]
=> [[1,5,8,12,13,14],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? => ? = 4 - 2
[6,2,2,2,1,1]
=> [[1,4,11,12,13,14],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? => ? = 6 - 2
[5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> 1100100010000 => ? => ? = 6 - 2
[5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> 1010100010000 => ? => ? = 6 - 2
[5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> 1010010010000 => ? => ? = 6 - 2
[6,5,1,1,1,1]
=> [[1,6,7,8,9,15],[2,11,12,13,14],[3],[4],[5],[10]]
=> ? => ? => ? = 5 - 2
[6,4,2,1,1,1]
=> [[1,5,8,9,14,15],[2,7,12,13],[3,11],[4],[6],[10]]
=> ? => ? => ? = 6 - 2
[6,3,2,2,1,1]
=> [[1,4,9,13,14,15],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? => ? => ? = 6 - 2
[6,2,2,2,2,1]
=> [[1,3,12,13,14,15],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 5 - 2
[5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> 10100100010000 => ? => ? = 6 - 2
[6,5,2,1,1,1]
=> [[1,5,8,9,10,16],[2,7,13,14,15],[3,12],[4],[6],[11]]
=> ? => ? => ? = 5 - 2
[6,4,3,1,1,1]
=> [[1,5,6,10,15,16],[2,8,9,14],[3,12,13],[4],[7],[11]]
=> ? => ? => ? = 6 - 2
[6,3,3,2,1,1]
=> [[1,4,7,14,15,16],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? => ? => ? = 6 - 2
[6,3,2,2,2,1]
=> [[1,3,10,14,15,16],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? => ? => ? = 5 - 2
[6,5,3,1,1,1]
=> [[1,5,6,10,11,17],[2,8,9,15,16],[3,13,14],[4],[7],[12]]
=> ? => ? => ? = 7 - 2
[6,5,2,2,1,1]
=> [[1,4,9,10,11,17],[2,6,14,15,16],[3,8],[5,13],[7],[12]]
=> ? => ? => ? = 7 - 2
[6,4,3,2,1,1]
=> [[1,4,7,11,16,17],[2,6,10,15],[3,9,14],[5,13],[8],[12]]
=> ? => ? => ? = 4 - 2
[6,4,2,2,2,1]
=> [[1,3,10,11,16,17],[2,5,14,15],[4,7],[6,9],[8,13],[12]]
=> ? => ? => ? = 7 - 2
[6,3,3,2,2,1]
=> [[1,3,8,15,16,17],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? => ? => ? = 7 - 2
[6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? => ? => ? = 7 - 2
[6,4,4,3,2,1]
=> [[1,3,6,10,19,20],[2,5,9,14],[4,8,13,18],[7,12,17],[11,16],[15]]
=> ? => ? => ? = 7 - 2
[6,5,3,3,2,1]
=> [[1,3,6,13,14,20],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
=> ? => ? => ? = 7 - 2
[6,4,3,3,2,1]
=> [[1,3,6,13,18,19],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
=> ? => ? => ? = 6 - 2
[6,3,3,3,2,1]
=> [[1,3,6,16,17,18],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? => ? => ? = 6 - 2
[6,5,4,2,2,1]
=> [[1,3,8,9,14,20],[2,5,12,13,19],[4,7,17,18],[6,11],[10,16],[15]]
=> ? => ? => ? = 7 - 2
[6,5,3,2,2,1]
=> [[1,3,8,12,13,19],[2,5,11,17,18],[4,7,16],[6,10],[9,15],[14]]
=> ? => ? => ? = 6 - 2
[6,4,3,2,2,1]
=> [[1,3,8,12,17,18],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
=> ? => ? => ? = 5 - 2
[6,5,2,2,2,1]
=> [[1,3,10,11,12,18],[2,5,15,16,17],[4,7],[6,9],[8,14],[13]]
=> ? => ? => ? = 6 - 2
[6,5,4,3,1,1]
=> [[1,4,5,9,14,20],[2,7,8,13,19],[3,11,12,18],[6,16,17],[10],[15]]
=> ? => ? => ? = 7 - 2
[6,4,4,3,1,1]
=> [[1,4,5,9,18,19],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
=> ? => ? => ? = 7 - 2
[6,5,4,2,1,1]
=> [[1,4,7,8,13,19],[2,6,11,12,18],[3,10,16,17],[5,15],[9],[14]]
=> ? => ? => ? = 6 - 2
[6,5,3,2,1,1]
=> [[1,4,7,11,12,18],[2,6,10,16,17],[3,9,15],[5,14],[8],[13]]
=> ? => ? => ? = 5 - 2
[6,5,4,1,1,1]
=> [[1,5,6,7,12,18],[2,9,10,11,17],[3,14,15,16],[4],[8],[13]]
=> ? => ? => ? = 6 - 2
Description
The length of the shortest palindromic decomposition of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
Matching statistic: St001020
Mp00321: Integer partitions ā2-conjugateā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001020: Dyck paths ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 80%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001020: Dyck paths ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 80%
Values
[2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 3 + 2
[3,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 6 = 4 + 2
[3,2,1]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 6 = 4 + 2
[4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 6 = 4 + 2
[4,2,1,1]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 4 + 2
[5,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 4 + 2
[4,3,1,1]
=> [5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> 7 = 5 + 2
[4,2,2,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]
=> ? = 5 + 2
[5,2,1,1,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 6 = 4 + 2
[4,3,2,1]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 5 + 2
[6,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 4 + 2
[5,3,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 2
[5,2,2,1,1]
=> [6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 6 + 2
[6,2,1,1,1,1]
=> [6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 4 + 2
[5,4,1,1,1]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> 7 = 5 + 2
[5,3,2,1,1]
=> [4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> 6 = 4 + 2
[5,2,2,2,1]
=> [7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 5 + 2
[6,3,1,1,1,1]
=> [5,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 6 + 2
[6,2,2,1,1,1]
=> [8,2,1,1,1]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> ? = 6 + 2
[5,4,2,1,1]
=> [6,3,3,1]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 5 + 2
[5,3,2,2,1]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> 7 = 5 + 2
[6,4,1,1,1,1]
=> [7,2,2,1,1,1]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 6 + 2
[6,3,2,1,1,1]
=> [6,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 + 2
[6,2,2,2,1,1]
=> [10,2,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 6 + 2
[5,4,3,1,1]
=> [5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> 8 = 6 + 2
[5,4,2,2,1]
=> [8,3,3]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 6 + 2
[5,3,3,2,1]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 6 + 2
[6,5,1,1,1,1]
=> [6,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 5 + 2
[6,4,2,1,1,1]
=> [9,2,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 6 + 2
[6,3,2,2,1,1]
=> [8,5,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 6 + 2
[6,2,2,2,2,1]
=> [11,2,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 5 + 2
[5,4,3,2,1]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 6 + 2
[6,5,2,1,1,1]
=> [7,6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 5 + 2
[6,4,3,1,1,1]
=> [7,3,2,2,1,1]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 6 + 2
[6,3,3,2,1,1]
=> [6,5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 6 + 2
[6,3,2,2,2,1]
=> [9,5,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,0,1,0]
=> ?
=> ? = 5 + 2
[6,5,3,1,1,1]
=> [6,3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 7 + 2
[6,5,2,2,1,1]
=> [8,5,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> ?
=> ? = 7 + 2
[6,4,3,2,1,1]
=> [8,7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 4 + 2
[6,4,2,2,2,1]
=> [11,4,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 7 + 2
[6,3,3,2,2,1]
=> [7,5,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 7 + 2
[6,5,4,3,2,1]
=> ?
=> ?
=> ?
=> ? = 7 + 2
[6,4,4,3,2,1]
=> ?
=> ?
=> ?
=> ? = 7 + 2
[6,5,3,3,2,1]
=> ?
=> ?
=> ?
=> ? = 7 + 2
[6,4,3,3,2,1]
=> ?
=> ?
=> ?
=> ? = 6 + 2
[6,3,3,3,2,1]
=> ?
=> ?
=> ?
=> ? = 6 + 2
[6,5,4,2,2,1]
=> ?
=> ?
=> ?
=> ? = 7 + 2
[6,5,3,2,2,1]
=> ?
=> ?
=> ?
=> ? = 6 + 2
[6,4,3,2,2,1]
=> ?
=> ?
=> ?
=> ? = 5 + 2
[6,5,2,2,2,1]
=> ?
=> ?
=> ?
=> ? = 6 + 2
[6,5,4,3,1,1]
=> ?
=> ?
=> ?
=> ? = 7 + 2
[6,4,4,3,1,1]
=> ?
=> ?
=> ?
=> ? = 7 + 2
[6,5,4,2,1,1]
=> ?
=> ?
=> ?
=> ? = 6 + 2
[6,5,3,2,1,1]
=> ?
=> ?
=> ?
=> ? = 5 + 2
[6,5,4,1,1,1]
=> ?
=> ?
=> ?
=> ? = 6 + 2
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001280
Mp00179: Integer partitions āto skew partitionā¶ Skew partitions
Mp00185: Skew partitions ācell posetā¶ Posets
Mp00110: Posets āGreene-Kleitman invariantā¶ Integer partitions
St001280: Integer partitions ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Mp00185: Skew partitions ācell posetā¶ Posets
Mp00110: Posets āGreene-Kleitman invariantā¶ Integer partitions
St001280: Integer partitions ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Values
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 1 = 3 - 2
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 2 = 4 - 2
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [3,2,1]
=> 2 = 4 - 2
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> [4,3]
=> 2 = 4 - 2
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> [4,3,1]
=> 2 = 4 - 2
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
=> [5,4]
=> 2 = 4 - 2
[4,3,1,1]
=> [[4,3,1,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> [4,3,2]
=> 3 = 5 - 2
[4,2,2,1]
=> [[4,2,2,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> [4,3,2]
=> 3 = 5 - 2
[5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(7,9),(8,4),(8,9)],10)
=> [5,4,1]
=> 2 = 4 - 2
[4,3,2,1]
=> [[4,3,2,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> [4,3,2,1]
=> 3 = 5 - 2
[6,1,1,1,1,1]
=> [[6,1,1,1,1,1],[]]
=> ?
=> ?
=> ? = 4 - 2
[5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ?
=> ? = 6 - 2
[5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ?
=> ? = 6 - 2
[6,2,1,1,1,1]
=> [[6,2,1,1,1,1],[]]
=> ?
=> ?
=> ? = 4 - 2
[5,4,1,1,1]
=> [[5,4,1,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ?
=> ? = 5 - 2
[5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11)],12)
=> ?
=> ? = 4 - 2
[5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ?
=> ? = 5 - 2
[6,3,1,1,1,1]
=> [[6,3,1,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,2,2,1,1,1]
=> [[6,2,2,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ?
=> ? = 5 - 2
[5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ?
=> ? = 5 - 2
[6,4,1,1,1,1]
=> [[6,4,1,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,2,1,1,1]
=> [[6,3,2,1,1,1],[]]
=> ?
=> ?
=> ? = 4 - 2
[6,2,2,2,1,1]
=> [[6,2,2,2,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ?
=> ? = 6 - 2
[5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(13,11)],14)
=> ?
=> ? = 6 - 2
[5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ?
=> ? = 6 - 2
[6,5,1,1,1,1]
=> [[6,5,1,1,1,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,4,2,1,1,1]
=> [[6,4,2,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,2,2,1,1]
=> [[6,3,2,2,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,2,2,2,2,1]
=> [[6,2,2,2,2,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(12,14),(13,11),(13,14)],15)
=> ?
=> ? = 6 - 2
[6,5,2,1,1,1]
=> [[6,5,2,1,1,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,4,3,1,1,1]
=> [[6,4,3,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,3,2,1,1]
=> [[6,3,3,2,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,2,2,2,1]
=> [[6,3,2,2,2,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,5,3,1,1,1]
=> [[6,5,3,1,1,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,2,2,1,1]
=> [[6,5,2,2,1,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,4,3,2,1,1]
=> [[6,4,3,2,1,1],[]]
=> ?
=> ?
=> ? = 4 - 2
[6,4,2,2,2,1]
=> [[6,4,2,2,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,3,3,2,2,1]
=> [[6,3,3,2,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,4,3,2,1]
=> [[6,5,4,3,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,4,4,3,2,1]
=> [[6,4,4,3,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,3,3,2,1]
=> [[6,5,3,3,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,4,3,3,2,1]
=> [[6,4,3,3,2,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,3,3,3,2,1]
=> [[6,3,3,3,2,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,5,4,2,2,1]
=> [[6,5,4,2,2,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,3,2,2,1]
=> [[6,5,3,2,2,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,4,3,2,2,1]
=> [[6,4,3,2,2,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,5,2,2,2,1]
=> [[6,5,2,2,2,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,5,4,3,1,1]
=> [[6,5,4,3,1,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,4,4,3,1,1]
=> [[6,4,4,3,1,1],[]]
=> ?
=> ?
=> ? = 7 - 2
[6,5,4,2,1,1]
=> [[6,5,4,2,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
[6,5,3,2,1,1]
=> [[6,5,3,2,1,1],[]]
=> ?
=> ?
=> ? = 5 - 2
[6,5,4,1,1,1]
=> [[6,5,4,1,1,1],[]]
=> ?
=> ?
=> ? = 6 - 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000291
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00134: Standard tableaux ādescent wordā¶ Binary words
Mp00234: Binary words āvalleys-to-peaksā¶ Binary words
St000291: Binary words ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Mp00134: Standard tableaux ādescent wordā¶ Binary words
Mp00234: Binary words āvalleys-to-peaksā¶ Binary words
St000291: Binary words ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Values
[2,1]
=> [[1,3],[2]]
=> 10 => 11 => 0 = 3 - 3
[3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1101 => 1 = 4 - 3
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 11001 => 1 = 4 - 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 111000 => 111001 => 1 = 4 - 3
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 1101000 => 1110001 => 1 = 4 - 3
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> 11110000 => 11110001 => 1 = 4 - 3
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> 11001000 => 11010001 => 2 = 5 - 3
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> 10101000 => 11010001 => 2 = 5 - 3
[5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> 111010000 => 111100001 => 1 = 4 - 3
[4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> 101001000 => 110010001 => 2 = 5 - 3
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 4 - 3
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> 1110010000 => ? => ? = 6 - 3
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> 1101010000 => ? => ? = 6 - 3
[6,2,1,1,1,1]
=> [[1,6,9,10,11,12],[2,8],[3],[4],[5],[7]]
=> ? => ? => ? = 4 - 3
[5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> 11100010000 => ? => ? = 5 - 3
[5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> 11010010000 => ? => ? = 4 - 3
[5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> 10101010000 => ? => ? = 5 - 3
[6,3,1,1,1,1]
=> [[1,6,7,11,12,13],[2,9,10],[3],[4],[5],[8]]
=> ? => ? => ? = 6 - 3
[6,2,2,1,1,1]
=> [[1,5,10,11,12,13],[2,7],[3,9],[4],[6],[8]]
=> ? => ? => ? = 6 - 3
[5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> 110100010000 => ? => ? = 5 - 3
[5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? => ? = 5 - 3
[6,4,1,1,1,1]
=> [[1,6,7,8,13,14],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? => ? = 6 - 3
[6,3,2,1,1,1]
=> [[1,5,8,12,13,14],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? => ? = 4 - 3
[6,2,2,2,1,1]
=> [[1,4,11,12,13,14],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? => ? = 6 - 3
[5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> 1100100010000 => ? => ? = 6 - 3
[5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> 1010100010000 => ? => ? = 6 - 3
[5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> 1010010010000 => ? => ? = 6 - 3
[6,5,1,1,1,1]
=> [[1,6,7,8,9,15],[2,11,12,13,14],[3],[4],[5],[10]]
=> ? => ? => ? = 5 - 3
[6,4,2,1,1,1]
=> [[1,5,8,9,14,15],[2,7,12,13],[3,11],[4],[6],[10]]
=> ? => ? => ? = 6 - 3
[6,3,2,2,1,1]
=> [[1,4,9,13,14,15],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? => ? => ? = 6 - 3
[6,2,2,2,2,1]
=> [[1,3,12,13,14,15],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 5 - 3
[5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> 10100100010000 => ? => ? = 6 - 3
[6,5,2,1,1,1]
=> [[1,5,8,9,10,16],[2,7,13,14,15],[3,12],[4],[6],[11]]
=> ? => ? => ? = 5 - 3
[6,4,3,1,1,1]
=> [[1,5,6,10,15,16],[2,8,9,14],[3,12,13],[4],[7],[11]]
=> ? => ? => ? = 6 - 3
[6,3,3,2,1,1]
=> [[1,4,7,14,15,16],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? => ? => ? = 6 - 3
[6,3,2,2,2,1]
=> [[1,3,10,14,15,16],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? => ? => ? = 5 - 3
[6,5,3,1,1,1]
=> [[1,5,6,10,11,17],[2,8,9,15,16],[3,13,14],[4],[7],[12]]
=> ? => ? => ? = 7 - 3
[6,5,2,2,1,1]
=> [[1,4,9,10,11,17],[2,6,14,15,16],[3,8],[5,13],[7],[12]]
=> ? => ? => ? = 7 - 3
[6,4,3,2,1,1]
=> [[1,4,7,11,16,17],[2,6,10,15],[3,9,14],[5,13],[8],[12]]
=> ? => ? => ? = 4 - 3
[6,4,2,2,2,1]
=> [[1,3,10,11,16,17],[2,5,14,15],[4,7],[6,9],[8,13],[12]]
=> ? => ? => ? = 7 - 3
[6,3,3,2,2,1]
=> [[1,3,8,15,16,17],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? => ? => ? = 7 - 3
[6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? => ? => ? = 7 - 3
[6,4,4,3,2,1]
=> [[1,3,6,10,19,20],[2,5,9,14],[4,8,13,18],[7,12,17],[11,16],[15]]
=> ? => ? => ? = 7 - 3
[6,5,3,3,2,1]
=> [[1,3,6,13,14,20],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
=> ? => ? => ? = 7 - 3
[6,4,3,3,2,1]
=> [[1,3,6,13,18,19],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
=> ? => ? => ? = 6 - 3
[6,3,3,3,2,1]
=> [[1,3,6,16,17,18],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? => ? => ? = 6 - 3
[6,5,4,2,2,1]
=> [[1,3,8,9,14,20],[2,5,12,13,19],[4,7,17,18],[6,11],[10,16],[15]]
=> ? => ? => ? = 7 - 3
[6,5,3,2,2,1]
=> [[1,3,8,12,13,19],[2,5,11,17,18],[4,7,16],[6,10],[9,15],[14]]
=> ? => ? => ? = 6 - 3
[6,4,3,2,2,1]
=> [[1,3,8,12,17,18],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
=> ? => ? => ? = 5 - 3
[6,5,2,2,2,1]
=> [[1,3,10,11,12,18],[2,5,15,16,17],[4,7],[6,9],[8,14],[13]]
=> ? => ? => ? = 6 - 3
[6,5,4,3,1,1]
=> [[1,4,5,9,14,20],[2,7,8,13,19],[3,11,12,18],[6,16,17],[10],[15]]
=> ? => ? => ? = 7 - 3
[6,4,4,3,1,1]
=> [[1,4,5,9,18,19],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
=> ? => ? => ? = 7 - 3
[6,5,4,2,1,1]
=> [[1,4,7,8,13,19],[2,6,11,12,18],[3,10,16,17],[5,15],[9],[14]]
=> ? => ? => ? = 6 - 3
[6,5,3,2,1,1]
=> [[1,4,7,11,12,18],[2,6,10,16,17],[3,9,15],[5,14],[8],[13]]
=> ? => ? => ? = 5 - 3
[6,5,4,1,1,1]
=> [[1,5,6,7,12,18],[2,9,10,11,17],[3,14,15,16],[4],[8],[13]]
=> ? => ? => ? = 6 - 3
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00134: Standard tableaux ādescent wordā¶ Binary words
Mp00280: Binary words āpath rowmotionā¶ Binary words
St000292: Binary words ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Mp00134: Standard tableaux ādescent wordā¶ Binary words
Mp00280: Binary words āpath rowmotionā¶ Binary words
St000292: Binary words ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 60%
Values
[2,1]
=> [[1,3],[2]]
=> 10 => 11 => 0 = 3 - 3
[3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0111 => 1 = 4 - 3
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 11001 => 1 = 4 - 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 111000 => 001111 => 1 = 4 - 3
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 1101000 => 1110001 => 1 = 4 - 3
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> 11110000 => 00011111 => 1 = 4 - 3
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> 11001000 => 01110001 => 2 = 5 - 3
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> 10101000 => 11010001 => 2 = 5 - 3
[5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> 111010000 => 111100001 => 1 = 4 - 3
[4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> 101001000 => 110010001 => 2 = 5 - 3
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? => ? = 4 - 3
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> 1110010000 => ? => ? = 6 - 3
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> 1101010000 => ? => ? = 6 - 3
[6,2,1,1,1,1]
=> [[1,6,9,10,11,12],[2,8],[3],[4],[5],[7]]
=> ? => ? => ? = 4 - 3
[5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> 11100010000 => ? => ? = 5 - 3
[5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> 11010010000 => ? => ? = 4 - 3
[5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> 10101010000 => ? => ? = 5 - 3
[6,3,1,1,1,1]
=> [[1,6,7,11,12,13],[2,9,10],[3],[4],[5],[8]]
=> ? => ? => ? = 6 - 3
[6,2,2,1,1,1]
=> [[1,5,10,11,12,13],[2,7],[3,9],[4],[6],[8]]
=> ? => ? => ? = 6 - 3
[5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> 110100010000 => ? => ? = 5 - 3
[5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? => ? = 5 - 3
[6,4,1,1,1,1]
=> [[1,6,7,8,13,14],[2,10,11,12],[3],[4],[5],[9]]
=> ? => ? => ? = 6 - 3
[6,3,2,1,1,1]
=> [[1,5,8,12,13,14],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? => ? = 4 - 3
[6,2,2,2,1,1]
=> [[1,4,11,12,13,14],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? => ? = 6 - 3
[5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> 1100100010000 => ? => ? = 6 - 3
[5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> 1010100010000 => ? => ? = 6 - 3
[5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> 1010010010000 => ? => ? = 6 - 3
[6,5,1,1,1,1]
=> [[1,6,7,8,9,15],[2,11,12,13,14],[3],[4],[5],[10]]
=> ? => ? => ? = 5 - 3
[6,4,2,1,1,1]
=> [[1,5,8,9,14,15],[2,7,12,13],[3,11],[4],[6],[10]]
=> ? => ? => ? = 6 - 3
[6,3,2,2,1,1]
=> [[1,4,9,13,14,15],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? => ? => ? = 6 - 3
[6,2,2,2,2,1]
=> [[1,3,12,13,14,15],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 5 - 3
[5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> 10100100010000 => ? => ? = 6 - 3
[6,5,2,1,1,1]
=> [[1,5,8,9,10,16],[2,7,13,14,15],[3,12],[4],[6],[11]]
=> ? => ? => ? = 5 - 3
[6,4,3,1,1,1]
=> [[1,5,6,10,15,16],[2,8,9,14],[3,12,13],[4],[7],[11]]
=> ? => ? => ? = 6 - 3
[6,3,3,2,1,1]
=> [[1,4,7,14,15,16],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? => ? => ? = 6 - 3
[6,3,2,2,2,1]
=> [[1,3,10,14,15,16],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? => ? => ? = 5 - 3
[6,5,3,1,1,1]
=> [[1,5,6,10,11,17],[2,8,9,15,16],[3,13,14],[4],[7],[12]]
=> ? => ? => ? = 7 - 3
[6,5,2,2,1,1]
=> [[1,4,9,10,11,17],[2,6,14,15,16],[3,8],[5,13],[7],[12]]
=> ? => ? => ? = 7 - 3
[6,4,3,2,1,1]
=> [[1,4,7,11,16,17],[2,6,10,15],[3,9,14],[5,13],[8],[12]]
=> ? => ? => ? = 4 - 3
[6,4,2,2,2,1]
=> [[1,3,10,11,16,17],[2,5,14,15],[4,7],[6,9],[8,13],[12]]
=> ? => ? => ? = 7 - 3
[6,3,3,2,2,1]
=> [[1,3,8,15,16,17],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? => ? => ? = 7 - 3
[6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? => ? => ? = 7 - 3
[6,4,4,3,2,1]
=> [[1,3,6,10,19,20],[2,5,9,14],[4,8,13,18],[7,12,17],[11,16],[15]]
=> ? => ? => ? = 7 - 3
[6,5,3,3,2,1]
=> [[1,3,6,13,14,20],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
=> ? => ? => ? = 7 - 3
[6,4,3,3,2,1]
=> [[1,3,6,13,18,19],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
=> ? => ? => ? = 6 - 3
[6,3,3,3,2,1]
=> [[1,3,6,16,17,18],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? => ? => ? = 6 - 3
[6,5,4,2,2,1]
=> [[1,3,8,9,14,20],[2,5,12,13,19],[4,7,17,18],[6,11],[10,16],[15]]
=> ? => ? => ? = 7 - 3
[6,5,3,2,2,1]
=> [[1,3,8,12,13,19],[2,5,11,17,18],[4,7,16],[6,10],[9,15],[14]]
=> ? => ? => ? = 6 - 3
[6,4,3,2,2,1]
=> [[1,3,8,12,17,18],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
=> ? => ? => ? = 5 - 3
[6,5,2,2,2,1]
=> [[1,3,10,11,12,18],[2,5,15,16,17],[4,7],[6,9],[8,14],[13]]
=> ? => ? => ? = 6 - 3
[6,5,4,3,1,1]
=> [[1,4,5,9,14,20],[2,7,8,13,19],[3,11,12,18],[6,16,17],[10],[15]]
=> ? => ? => ? = 7 - 3
[6,4,4,3,1,1]
=> [[1,4,5,9,18,19],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
=> ? => ? => ? = 7 - 3
[6,5,4,2,1,1]
=> [[1,4,7,8,13,19],[2,6,11,12,18],[3,10,16,17],[5,15],[9],[14]]
=> ? => ? => ? = 6 - 3
[6,5,3,2,1,1]
=> [[1,4,7,11,12,18],[2,6,10,16,17],[3,9,15],[5,14],[8],[13]]
=> ? => ? => ? = 5 - 3
[6,5,4,1,1,1]
=> [[1,5,6,7,12,18],[2,9,10,11,17],[3,14,15,16],[4],[8],[13]]
=> ? => ? => ? = 6 - 3
Description
The number of ascents of a binary word.
Matching statistic: St001040
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
St001040: Perfect matchings ā¶ ā¤Result quality: 16% āvalues known / values provided: 16%ādistinct values known / distinct values provided: 60%
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
St001040: Perfect matchings ā¶ ā¤Result quality: 16% āvalues known / values provided: 16%ādistinct values known / distinct values provided: 60%
Values
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 4 = 3 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 5 = 4 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 5 = 4 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 4 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 5 = 4 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> 5 = 4 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 6 = 5 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 6 = 5 + 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 4 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 6 = 5 + 1
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,14),(8,9),(10,13),(11,12)]
=> ? = 4 + 1
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)]
=> ? = 6 + 1
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]
=> ? = 6 + 1
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,14),(8,9),(10,13),(11,12)]
=> ? = 4 + 1
[5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 5 + 1
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [(1,12),(2,7),(3,6),(4,5),(8,11),(9,10)]
=> ? = 4 + 1
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> ? = 5 + 1
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,9),(10,13),(11,12)]
=> ? = 6 + 1
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,9),(10,13),(11,12)]
=> ? = 6 + 1
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [(1,12),(2,11),(3,6),(4,5),(7,10),(8,9)]
=> ? = 5 + 1
[5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]
=> ? = 5 + 1
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,14),(4,5),(6,9),(7,8),(10,13),(11,12)]
=> ? = 6 + 1
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,14),(8,9),(10,13),(11,12)]
=> ? = 4 + 1
[6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,9),(7,8),(10,13),(11,12)]
=> ? = 6 + 1
[5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)]
=> ? = 6 + 1
[5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]
=> ? = 6 + 1
[5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
=> ? = 6 + 1
[6,5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [(1,14),(2,5),(3,4),(6,9),(7,8),(10,13),(11,12)]
=> ? = 6 + 1
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,14),(6,9),(7,8),(10,13),(11,12)]
=> ? = 6 + 1
[6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ? = 5 + 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> ? = 6 + 1
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[6,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [(1,14),(2,9),(3,4),(5,8),(6,7),(10,13),(11,12)]
=> ? = 6 + 1
[6,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [(1,2),(3,14),(4,9),(5,8),(6,7),(10,13),(11,12)]
=> ? = 6 + 1
[6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ? = 5 + 1
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ? = 7 + 1
[6,5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ? = 7 + 1
[6,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [(1,14),(2,9),(3,8),(4,7),(5,6),(10,13),(11,12)]
=> ? = 4 + 1
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,7),(5,6),(8,13),(9,12),(10,11)]
=> ? = 7 + 1
[6,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,7),(5,6),(8,13),(9,12),(10,11)]
=> ? = 7 + 1
[6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 7 + 1
[6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,14),(2,3),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 7 + 1
[6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> ? = 7 + 1
[6,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,5),(3,4),(6,13),(7,12),(8,11),(9,10)]
=> ? = 6 + 1
[6,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 6 + 1
[6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> ? = 7 + 1
[6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 6 + 1
[6,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [(1,14),(2,7),(3,6),(4,5),(8,13),(9,12),(10,11)]
=> ? = 5 + 1
[6,5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ? = 6 + 1
[6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 7 + 1
[6,4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> ? = 7 + 1
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,7),(5,6),(8,11),(9,10)]
=> ? = 6 + 1
[6,5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [(1,14),(2,13),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 5 + 1
[6,5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ? = 6 + 1
Description
The depth of the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St000150
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00132: Dyck paths āswitch returns and last double riseā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000150: Integer partitions ā¶ ā¤Result quality: 16% āvalues known / values provided: 16%ādistinct values known / distinct values provided: 60%
Mp00132: Dyck paths āswitch returns and last double riseā¶ Dyck paths
Mp00027: Dyck paths āto partitionā¶ Integer partitions
St000150: Integer partitions ā¶ ā¤Result quality: 16% āvalues known / values provided: 16%ādistinct values known / distinct values provided: 60%
Values
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 0 = 3 - 3
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1 = 4 - 3
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1 = 4 - 3
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 1 = 4 - 3
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> 1 = 4 - 3
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,3,2,1]
=> 1 = 4 - 3
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 5 - 3
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,1,1,1]
=> 2 = 5 - 3
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,3,3,2,1]
=> 1 = 4 - 3
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1,1,1]
=> 2 = 5 - 3
[6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,8,7,6,5,4,4,3,2,1]
=> ? = 4 - 3
[5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [7,6,5,5,3,3,2,1]
=> ? = 6 - 3
[5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [7,6,3,2,2,2,2,1]
=> ? = 6 - 3
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 4 - 3
[5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,3,3,2,1]
=> ? = 5 - 3
[5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,3,2,2,2,1]
=> ? = 4 - 3
[5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,4,3,2,2,2,2,1]
=> ? = 5 - 3
[6,3,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,2,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [7,6,6,3,3,2,1,1]
=> ? = 5 - 3
[5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,3,2,2,2,1]
=> ? = 5 - 3
[6,4,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,3,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 4 - 3
[6,2,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [7,6,6,2,2,2,1,1]
=> ? = 6 - 3
[5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,4,4,3,2,2,1]
=> ? = 6 - 3
[5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,2,2,2,2,1]
=> ? = 6 - 3
[6,5,1,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 5 - 3
[6,4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,3,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,2,2,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 5 - 3
[5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [7,4,4,4,2,2,2,1]
=> ? = 6 - 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 5 - 3
[6,4,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,3,3,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,3,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 5 - 3
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,5,2,2,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,4,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 4 - 3
[6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,3,3,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ?
=> ? = 7 - 3
[6,4,4,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,5,3,3,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,4,3,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,5,4,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,5,3,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,4,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 5 - 3
[6,5,2,2,2,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,4,4,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 7 - 3
[6,5,4,2,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
[6,5,3,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 5 - 3
[6,5,4,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,0]
=> ?
=> ?
=> ? = 6 - 3
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
The following 190 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St000836The number of descents of distance 2 of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000039The number of crossings of a permutation. St000673The number of non-fixed points of a permutation. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001471The magnitude of a Dyck path. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000331The number of upper interactions of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000837The number of ascents of distance 2 of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001424The number of distinct squares in a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St000024The number of double up and double down steps of a Dyck path. St000257The number of distinct parts of a partition that occur at least twice. St000442The maximal area to the right of an up step of a Dyck path. St000480The number of lower covers of a partition in dominance order. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001584The area statistic between a Dyck path and its bounce path. St001729The number of visible descents of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000144The pyramid weight of the Dyck path. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St000015The number of peaks 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 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001566The length of the longest arithmetic progression in a permutation. St000665The number of rafts of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000871The number of very big ascents of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000489The number of cycles of a permutation of length at most 3. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000019The cardinality of the support of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000035The number of left outer peaks of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000495The number of inversions of distance at most 2 of a permutation. St000670The reversal length of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000834The number of right outer peaks of a permutation. St000920The logarithmic height of a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001405The number of bonds in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000353The number of inner valleys of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000441The number of successions of a permutation. St000538The number of even inversions of a permutation. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001114The number of odd descents of a permutation. St001469The holeyness of a permutation. St001728The number of invisible descents of a permutation. St001712The number of natural descents of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000226The convexity of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001330The hat guessing number of a graph. St000141The maximum drop size of a permutation. St000201The number of leaf nodes in a binary tree. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000652The maximal difference between successive positions of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001807The lower middle entry of a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000052The number of valleys of a Dyck path not on the x-axis. St000306The bounce count of a Dyck path. St000335The difference of lower and upper interactions. St000624The normalized sum of the minimal distances to a greater element. St000742The number of big ascents of a permutation after prepending zero. St000862The number of parts of the shifted shape of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001115The number of even descents of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001252Half the sum of the even parts of a partition. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001587Half of the largest even part of an integer partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000057The Shynar inversion number of a standard tableau. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000237The number of small exceedances. St000352The Elizalde-Pak rank of a permutation. St000365The number of double ascents of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000884The number of isolated descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001335The cardinality of a minimal cycle-isolating set of a graph. St001394The genus of a permutation. St001470The cyclic holeyness of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001695The natural comajor index of a standard Young tableau. St001859The number of factors of the Stanley symmetric function associated with a permutation.
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