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Matching statistic: St000198
St000198: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 2
[.,[.,.]]
=> 10
[[.,.],.]
=> 12
[.,[.,[.,.]]]
=> 42
[.,[[.,.],.]]
=> 44
[[.,.],[.,.]]
=> 50
[[.,[.,.]],.]
=> 52
[[[.,.],.],.]
=> 56
[.,[.,[.,[.,.]]]]
=> 170
[.,[.,[[.,.],.]]]
=> 172
[.,[[.,.],[.,.]]]
=> 178
[.,[[.,[.,.]],.]]
=> 180
[.,[[[.,.],.],.]]
=> 184
[[.,.],[.,[.,.]]]
=> 202
[[.,.],[[.,.],.]]
=> 204
[[.,[.,.]],[.,.]]
=> 210
[[[.,.],.],[.,.]]
=> 226
[[.,[.,[.,.]]],.]
=> 212
[[.,[[.,.],.]],.]
=> 216
[[[.,.],[.,.]],.]
=> 228
[[[.,[.,.]],.],.]
=> 232
[[[[.,.],.],.],.]
=> 240
[.,[.,[.,[.,[.,.]]]]]
=> 682
[.,[.,[.,[[.,.],.]]]]
=> 684
[.,[.,[[.,.],[.,.]]]]
=> 690
[.,[.,[[.,[.,.]],.]]]
=> 692
[.,[.,[[[.,.],.],.]]]
=> 696
[.,[[.,.],[.,[.,.]]]]
=> 714
[.,[[.,.],[[.,.],.]]]
=> 716
[.,[[.,[.,.]],[.,.]]]
=> 722
[.,[[[.,.],.],[.,.]]]
=> 738
[.,[[.,[.,[.,.]]],.]]
=> 724
[.,[[.,[[.,.],.]],.]]
=> 728
[.,[[[.,.],[.,.]],.]]
=> 740
[.,[[[.,[.,.]],.],.]]
=> 744
[.,[[[[.,.],.],.],.]]
=> 752
[[.,.],[.,[.,[.,.]]]]
=> 810
[[.,.],[.,[[.,.],.]]]
=> 812
[[.,.],[[.,.],[.,.]]]
=> 818
[[.,.],[[.,[.,.]],.]]
=> 820
[[.,.],[[[.,.],.],.]]
=> 824
[[.,[.,.]],[.,[.,.]]]
=> 842
[[.,[.,.]],[[.,.],.]]
=> 844
[[[.,.],.],[.,[.,.]]]
=> 906
[[[.,.],.],[[.,.],.]]
=> 908
[[.,[.,[.,.]]],[.,.]]
=> 850
[[.,[[.,.],.]],[.,.]]
=> 866
[[[.,.],[.,.]],[.,.]]
=> 914
[[[.,[.,.]],.],[.,.]]
=> 930
[[[[.,.],.],.],[.,.]]
=> 962
Description
A decimal representation of a binary tree as a code word.
Code word constructed by mapping a binary tree to its corresponding set of matching parentheses and then replacing every '''''(''''' with a $1$ and every ''''')''''' with a $0$. We then convert the resulting binary code word to a decimal number.
Matching statistic: St000289
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000289: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Mp00093: Dyck paths —to binary word⟶ Binary words
St000289: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Values
[.,.]
=> [1,0]
=> 10 => 2
[.,[.,.]]
=> [1,0,1,0]
=> 1010 => 10
[[.,.],.]
=> [1,1,0,0]
=> 1100 => 12
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 101010 => 42
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 101100 => 44
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 110010 => 50
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 110100 => 52
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 111000 => 56
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 170
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 172
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 178
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 180
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 184
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 202
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 204
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 210
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 226
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 212
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 216
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 228
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 232
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 240
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 682
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 684
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 690
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 692
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 696
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 714
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? = 716
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => ? = 722
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 738
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 724
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 728
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 740
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 744
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 752
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 810
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? = 812
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? = 818
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? = 820
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 824
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => ? = 842
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => ? = 844
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? = 906
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 908
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => ? = 850
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 866
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 914
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 930
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 962
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 852
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => ? = 856
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => ? = 868
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => ? = 872
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => ? = 880
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 916
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => ? = 920
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? = 932
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? = 964
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? = 936
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? = 944
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => ? = 968
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => ? = 976
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? = 992
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 2730
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 2732
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => ? = 2738
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2740
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => ? = 2744
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => ? = 2762
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => ? = 2764
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 101011010010 => ? = 2770
Description
The decimal representation of a binary word.
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