Your data matches 5 different statistics following compositions of up to 3 maps.
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Mp00051: Ordered trees to Dyck pathDyck paths
St001202: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
Description
Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. Associate to this special CNakayama algebra a Dyck path as follows: In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra to which we can associate a Dyck path as the top boundary of the Auslander-Reiten quiver of the LNakayama algebra. The statistic gives half the dominant dimension of hte first indecomposable projective module in the special CNakayama algebra.
Matching statistic: St001184
Mp00048: Ordered trees left-right symmetryOrdered trees
Mp00051: Ordered trees to Dyck pathDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [1,0]
=> [1,0]
=> 1
[[],[]]
=> [[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[[]],[]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[],[]]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[[]]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[],[[]],[]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[]],[],[]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[]],[[]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[[[]]],[]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[[],[],[]]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[[]]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[[],[]]]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001355
Mp00139: Ordered trees Zeilberger's Strahler bijectionBinary trees
Mp00141: Binary trees pruning number to logarithmic heightDyck paths
Mp00093: Dyck paths to binary wordBinary words
St001355: Binary words ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> [1,0]
=> 10 => 1
[[],[]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1010 => 2
[[[]]]
=> [[.,.],.]
=> [1,1,0,0]
=> 1100 => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 101010 => 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 101100 => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 110010 => 2
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 111000 => 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 110100 => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 3
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 3
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 3
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 2
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 1
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 4
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 4
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 3
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 3
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 4
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? = 3
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 2
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => ? = 3
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 3
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 2
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 2
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 4
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? = 3
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? = 3
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => ? = 2
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? = 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => ? = 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? = 1
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => ? = 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? = 3
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? = 1
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => ? = 1
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => ? = 1
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => ? = 2
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? = 2
[[[],[],[[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 2
[[[],[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
[[[],[[],[]]]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 2
[[[],[[[]]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 1
[[[[]],[],[]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 2
[[[[]],[[]]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? = 1
[[[[],[]],[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? = 2
[[[[[]]],[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => ? = 1
[[[[],[],[]]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1
[[[[],[[]]]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => ? = 1
[[[[[]],[]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => ? = 1
[[[[[],[]]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => ? = 1
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 1
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 6
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 5
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => ? = 5
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => ? = 4
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 4
[[],[],[[]],[],[]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => ? = 5
[[],[],[[]],[[]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => ? = 4
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => ? = 3
Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Matching statistic: St001462
Mp00139: Ordered trees Zeilberger's Strahler bijectionBinary trees
Mp00141: Binary trees pruning number to logarithmic heightDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> [1,0]
=> [[1],[2]]
=> 1
[[],[]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[[]]]
=> [[.,.],.]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 2
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 3
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 3
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 3
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 1
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 4
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> ? = 4
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 3
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 3
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> ? = 4
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> ? = 3
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 2
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> ? = 3
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> ? = 3
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 2
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> ? = 4
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> ? = 3
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> ? = 3
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[1,2,3,6,7],[4,5,8,9,10]]
=> ? = 2
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> ? = 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> ? = 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> ? = 1
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> ? = 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> ? = 3
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> ? = 1
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> ? = 1
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> ? = 1
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> ? = 2
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> ? = 2
[[[],[],[[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 2
[[[],[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1
[[[],[[],[]]]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 2
[[[],[[[]]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> ? = 1
[[[[]],[],[]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 2
[[[[]],[[]]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> ? = 1
[[[[],[]],[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2
[[[[[]]],[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> ? = 1
[[[[],[],[]]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1
[[[[],[[]]]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> ? = 1
[[[[[]],[]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> ? = 1
[[[[[],[]]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> ? = 1
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> ? = 1
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> ? = 6
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> ? = 5
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> ? = 5
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> ? = 4
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 4
[[],[],[[]],[],[]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> ? = 5
[[],[],[[]],[[]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,6,9,10],[2,4,7,8,11,12]]
=> ? = 4
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> ? = 3
Description
The number of factors of a standard tableaux under concatenation. The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10]. This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St001553
Mp00139: Ordered trees Zeilberger's Strahler bijectionBinary trees
Mp00141: Binary trees pruning number to logarithmic heightDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001553: Dyck paths ⟶ ℤResult quality: 11% values known / values provided: 11%distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> [1,0]
=> [1,1,0,0]
=> 1
[[],[]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[[]]]
=> [[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 4
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 4
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 4
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 3
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 3
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 3
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 3
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 3
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 1
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[[],[],[[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[[[],[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[[],[[],[]]]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 2
[[[],[[[]]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 1
[[[[]],[],[]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[[[[]],[[]]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1
[[[[],[]],[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[[[[]]],[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[[[],[],[]]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[[[[],[[]]]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 1
[[[[[]],[]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[[[[[],[]]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 1
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 5
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 5
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 4
[[],[],[[]],[],[]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 5
[[],[],[[]],[[]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 4
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. The statistic returns zero in case that bimodule is the zero module.