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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001183
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001183: 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]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 3
Description
The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 83%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> 10 => 00 => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> 1010 => 0010 => 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> 1100 => 0100 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 101010 => 001010 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 010100 => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 010010 => 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 001100 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 111000 => 011000 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 00101010 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 01010100 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01010010 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00110100 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 01101000 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01001010 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 01100100 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 00110010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 00101100 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 01011000 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01100010 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 01001100 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00111000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 01110000 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0010101010 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0101010100 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0101010010 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0011010100 => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 0110101000 => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0101001010 => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 0110100100 => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0011010010 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0010110100 => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0101101000 => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0110100010 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0011100100 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101000 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0111010000 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0100101010 => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0110010100 => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 0110010010 => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0100110100 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 0111001000 => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0011001010 => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0101100100 => ? = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0010110010 => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0010101100 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0101011000 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0101100010 => ? = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0101001100 => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011011000 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0101110000 => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0110001010 => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 0111000100 => ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 0100110010 => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0100101100 => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 0110011000 => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0011100010 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0011001100 => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0010111000 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0110110000 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 0111000010 => ? = 3 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 0110001100 => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0100111000 => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0011110000 => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0111100000 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 001010101010 => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => 010101010100 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 110101010010 => 010101010010 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => 001101010100 => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 111010101000 => 011010101000 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 110101001010 => 010101001010 => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 111010100100 => 011010100100 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 101101010010 => 001101010010 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => 001011010100 => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 110110101000 => 010110101000 => ? = 5 - 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 83%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> 10 => 00 => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> 1010 => 0010 => 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> 1100 => 0100 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 101010 => 001010 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 010100 => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 010010 => 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 001100 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 111000 => 011000 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 00101010 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 01010100 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01010010 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00110100 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 01101000 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01001010 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 01100100 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 00110010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 00101100 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 01011000 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01100010 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 01001100 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00111000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 01110000 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0010101010 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0101010100 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0101010010 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0011010100 => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 0110101000 => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0101001010 => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 0110100100 => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0011010010 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0010110100 => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0101101000 => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0110100010 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0011100100 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101000 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0111010000 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0100101010 => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0110010100 => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 0110010010 => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0100110100 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 0111001000 => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0011001010 => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0101100100 => ? = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0010110010 => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0010101100 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0101011000 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0101100010 => ? = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0101001100 => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011011000 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0101110000 => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0110001010 => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 0111000100 => ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 0100110010 => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0100101100 => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 0110011000 => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0011100010 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0011001100 => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0010111000 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0110110000 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 0111000010 => ? = 3 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 0110001100 => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0100111000 => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0011110000 => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0111100000 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 001010101010 => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => 010101010100 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 110101010010 => 010101010010 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => 001101010100 => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 111010101000 => 011010101000 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 110101001010 => 010101001010 => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 111010100100 => 011010100100 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 101101010010 => 001101010010 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => 001011010100 => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 110110101000 => 010110101000 => ? = 5 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 83%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> 10 => 00 => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> 1010 => 0010 => 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> 1100 => 0100 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 101010 => 001010 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 010100 => 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 010010 => 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 001100 => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 111000 => 011000 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 00101010 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 01010100 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01010010 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00110100 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 01101000 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01001010 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 01100100 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 00110010 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 00101100 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 01011000 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01100010 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 01001100 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00111000 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 01110000 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0010101010 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0101010100 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 0101010010 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0011010100 => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 0110101000 => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 0101001010 => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 0110100100 => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0011010010 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0010110100 => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0101101000 => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 0110100010 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0011100100 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101000 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0111010000 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 0100101010 => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0110010100 => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 0110010010 => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0100110100 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 0111001000 => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0011001010 => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0101100100 => ? = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0010110010 => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0010101100 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0101011000 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 0101100010 => ? = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0101001100 => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011011000 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0101110000 => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 0110001010 => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 0111000100 => ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 0100110010 => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0100101100 => ? = 4 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 0110011000 => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0011100010 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0011001100 => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0010111000 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0110110000 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 0111000010 => ? = 3 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 0110001100 => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0100111000 => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0011110000 => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0111100000 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 001010101010 => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => 010101010100 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 110101010010 => 010101010010 => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => 001101010100 => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 111010101000 => 011010101000 => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 110101001010 => 010101001010 => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 111010100100 => 011010100100 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 101101010010 => 001101010010 => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => 001011010100 => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 110110101000 => 010110101000 => ? = 5 - 1
Description
The number of runs of ones in a binary word.
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