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Identifier
Values
=>
0=>0 1=>1 00=>0 01=>1 10=>1 11=>1 000=>0 001=>1 010=>1 011=>1 100=>1 101=>2 110=>2 111=>2 0000=>0 0001=>1 0010=>1 0011=>1 0100=>1 0101=>2 0110=>2 0111=>2 1000=>1 1001=>2 1010=>2 1011=>2 1100=>2 1101=>2 1110=>2 1111=>2 00000=>0 00001=>1 00010=>1 00011=>1 00100=>1 00101=>2 00110=>2 00111=>2 01000=>1 01001=>2 01010=>2 01011=>2 01100=>2 01101=>2 01110=>2 01111=>2 10000=>1 10001=>2 10010=>2 10011=>2 10100=>2 10101=>3 10110=>3 10111=>3 11000=>2 11001=>3 11010=>3 11011=>3 11100=>2 11101=>3 11110=>3 11111=>3 000000=>0 000001=>1 000010=>1 000011=>1 000100=>1 000101=>2 000110=>2 000111=>2 001000=>1 001001=>2 001010=>2 001011=>2 001100=>2 001101=>2 001110=>2 001111=>2 010000=>1 010001=>2 010010=>2 010011=>2 010100=>2 010101=>3 010110=>3 010111=>3 011000=>2 011001=>3 011010=>3 011011=>3 011100=>2 011101=>3 011110=>3 011111=>3 100000=>1 100001=>2 100010=>2 100011=>2 100100=>2 100101=>3 100110=>3 100111=>3 101000=>2 101001=>3 101010=>3 101011=>3 101100=>3 101101=>3 101110=>3 101111=>3 110000=>2 110001=>3 110010=>3 110011=>3 110100=>3 110101=>3 110110=>3 110111=>3 111000=>2 111001=>3 111010=>3 111011=>3 111100=>3 111101=>3 111110=>3 111111=>3 0000000=>0 0000001=>1 0000010=>1 0000011=>1 0000100=>1 0000101=>2 0000110=>2 0000111=>2 0001000=>1 0001001=>2 0001010=>2 0001011=>2 0001100=>2 0001101=>2 0001110=>2 0001111=>2 0010000=>1 0010001=>2 0010010=>2 0010011=>2 0010100=>2 0010101=>3 0010110=>3 0010111=>3 0011000=>2 0011001=>3 0011010=>3 0011011=>3 0011100=>2 0011101=>3 0011110=>3 0011111=>3 0100000=>1 0100001=>2 0100010=>2 0100011=>2 0100100=>2 0100101=>3 0100110=>3 0100111=>3 0101000=>2 0101001=>3 0101010=>3 0101011=>3 0101100=>3 0101101=>3 0101110=>3 0101111=>3 0110000=>2 0110001=>3 0110010=>3 0110011=>3 0110100=>3 0110101=>3 0110110=>3 0110111=>3 0111000=>2 0111001=>3 0111010=>3 0111011=>3 0111100=>3 0111101=>3 0111110=>3 0111111=>3 1000000=>1 1000001=>2 1000010=>2 1000011=>2 1000100=>2 1000101=>3 1000110=>3 1000111=>3 1001000=>2 1001001=>3 1001010=>3 1001011=>3 1001100=>3 1001101=>3 1001110=>3 1001111=>3 1010000=>2 1010001=>3 1010010=>3 1010011=>3 1010100=>3 1010101=>4 1010110=>4 1010111=>4 1011000=>3 1011001=>4 1011010=>4 1011011=>4 1011100=>3 1011101=>4 1011110=>4 1011111=>4 1100000=>2 1100001=>3 1100010=>3 1100011=>3 1100100=>3 1100101=>4 1100110=>4 1100111=>4 1101000=>3 1101001=>4 1101010=>4 1101011=>4 1101100=>3 1101101=>4 1101110=>4 1101111=>4 1110000=>2 1110001=>3 1110010=>3 1110011=>3 1110100=>3 1110101=>4 1110110=>4 1110111=>4 1111000=>3 1111001=>4 1111010=>4 1111011=>4 1111100=>3 1111101=>4 1111110=>4 1111111=>4 00000000=>0 00000001=>1 00000010=>1 00000011=>1 00000100=>1 00000101=>2 00000110=>2 00000111=>2 00001000=>1 00001001=>2 00001010=>2 00001011=>2 00001100=>2 00001101=>2 00001110=>2 00001111=>2 00010000=>1 00010001=>2 00010010=>2 00010011=>2 00010100=>2 00010101=>3 00010110=>3 00010111=>3 00011000=>2 00011001=>3 00011010=>3 00011011=>3 00011100=>2 00011101=>3 00011110=>3 00011111=>3 00100000=>1 00100001=>2 00100010=>2 00100011=>2 00100100=>2 00100101=>3 00100110=>3 00100111=>3 00101000=>2 00101001=>3 00101010=>3 00101011=>3 00101100=>3 00101101=>3 00101110=>3 00101111=>3 00110000=>2 00110001=>3 00110010=>3 00110011=>3 00110100=>3 00110101=>3 00110110=>3 00110111=>3 00111000=>2 00111001=>3 00111010=>3 00111011=>3 00111100=>3 00111101=>3 00111110=>3 00111111=>3 01000000=>1 01000001=>2 01000010=>2 01000011=>2 01000100=>2 01000101=>3 01000110=>3 01000111=>3 01001000=>2 01001001=>3 01001010=>3 01001011=>3 01001100=>3 01001101=>3 01001110=>3 01001111=>3 01010000=>2 01010001=>3 01010010=>3 01010011=>3 01010100=>3 01010101=>4 01010110=>4 01010111=>4 01011000=>3 01011001=>4 01011010=>4 01011011=>4 01011100=>3 01011101=>4 01011110=>4 01011111=>4 01100000=>2 01100001=>3 01100010=>3 01100011=>3 01100100=>3 01100101=>4 01100110=>4 01100111=>4 01101000=>3 01101001=>4 01101010=>4 01101011=>4 01101100=>3 01101101=>4 01101110=>4 01101111=>4 01110000=>2 01110001=>3 01110010=>3 01110011=>3 01110100=>3 01110101=>4 01110110=>4 01110111=>4 01111000=>3 01111001=>4 01111010=>4 01111011=>4 01111100=>3 01111101=>4 01111110=>4 01111111=>4 10000000=>1 10000001=>2 10000010=>2 10000011=>2 10000100=>2 10000101=>3 10000110=>3 10000111=>3 10001000=>2 10001001=>3 10001010=>3 10001011=>3 10001100=>3 10001101=>3 10001110=>3 10001111=>3 10010000=>2 10010001=>3 10010010=>3 10010011=>3 10010100=>3 10010101=>4 10010110=>4 10010111=>4 10011000=>3 10011001=>4 10011010=>4 10011011=>4 10011100=>3 10011101=>4 10011110=>4 10011111=>4 10100000=>2 10100001=>3 10100010=>3 10100011=>3 10100100=>3 10100101=>4 10100110=>4 10100111=>4 10101000=>3 10101001=>4 10101010=>4 10101011=>4 10101100=>4 10101101=>4 10101110=>4 10101111=>4 10110000=>3 10110001=>4 10110010=>4 10110011=>4 10110100=>4 10110101=>4 10110110=>4 10110111=>4 10111000=>3 10111001=>4 10111010=>4 10111011=>4 10111100=>4 10111101=>4 10111110=>4 10111111=>4 11000000=>2 11000001=>3 11000010=>3 11000011=>3 11000100=>3 11000101=>4 11000110=>4 11000111=>4 11001000=>3 11001001=>4 11001010=>4 11001011=>4 11001100=>4 11001101=>4 11001110=>4 11001111=>4 11010000=>3 11010001=>4 11010010=>4 11010011=>4 11010100=>4 11010101=>4 11010110=>4 11010111=>4 11011000=>3 11011001=>4 11011010=>4 11011011=>4 11011100=>4 11011101=>4 11011110=>4 11011111=>4 11100000=>2 11100001=>3 11100010=>3 11100011=>3 11100100=>3 11100101=>4 11100110=>4 11100111=>4 11101000=>3 11101001=>4 11101010=>4 11101011=>4 11101100=>4 11101101=>4 11101110=>4 11101111=>4 11110000=>3 11110001=>4 11110010=>4 11110011=>4 11110100=>4 11110101=>4 11110110=>4 11110111=>4 11111000=>3 11111001=>4 11111010=>4 11111011=>4 11111100=>4 11111101=>4 11111110=>4 11111111=>4 000000000=>0 000000001=>1 000000010=>1 000000011=>1 000000100=>1 000000101=>2 000000110=>2 000000111=>2 000001000=>1 000001001=>2 000001010=>2 000001011=>2 000001100=>2 000001101=>2 000001110=>2 000001111=>2 000010000=>1 000010001=>2 000010010=>2 000010011=>2 000010100=>2 000010101=>3 000010110=>3 000010111=>3 000011000=>2 000011001=>3 000011010=>3 000011011=>3 000011100=>2 000011101=>3 000011110=>3 000011111=>3 000100000=>1 000100001=>2 000100010=>2 000100011=>2 000100100=>2 000100101=>3 000100110=>3 000100111=>3 000101000=>2 000101001=>3 000101010=>3 000101011=>3 000101100=>3 000101101=>3 000101110=>3 000101111=>3 000110000=>2 000110001=>3 000110010=>3 000110011=>3 000110100=>3 000110101=>3 000110110=>3 000110111=>3 000111000=>2 000111001=>3 000111010=>3 000111011=>3 000111100=>3 000111101=>3 000111110=>3 000111111=>3 001000000=>1 001000001=>2 001000010=>2 001000011=>2 001000100=>2 001000101=>3 001000110=>3 001000111=>3 001001000=>2 001001001=>3 001001010=>3 001001011=>3 001001100=>3 001001101=>3 001001110=>3 001001111=>3 001010000=>2 001010001=>3 001010010=>3 001010011=>3 001010100=>3 001010101=>4 001010110=>4 001010111=>4 001011000=>3 001011001=>4 001011010=>4 001011011=>4 001011100=>3 001011101=>4 001011110=>4 001011111=>4 001100000=>2 001100001=>3 001100010=>3 001100011=>3 001100100=>3 001100101=>4 001100110=>4 001100111=>4 001101000=>3 001101001=>4 001101010=>4 001101011=>4 001101100=>3 001101101=>4 001101110=>4 001101111=>4 001110000=>2 001110001=>3 001110010=>3 001110011=>3 001110100=>3 001110101=>4 001110110=>4 001110111=>4 001111000=>3 001111001=>4 001111010=>4 001111011=>4 001111100=>3 001111101=>4 001111110=>4 001111111=>4 010000000=>1 010000001=>2 010000010=>2 010000011=>2 010000100=>2 010000101=>3 010000110=>3 010000111=>3 010001000=>2 010001001=>3 010001010=>3 010001011=>3 010001100=>3 010001101=>3 010001110=>3 010001111=>3 010010000=>2 010010001=>3 010010010=>3 010010011=>3 010010100=>3 010010101=>4 010010110=>4 010010111=>4 010011000=>3 010011001=>4 010011010=>4 010011011=>4 010011100=>3 010011101=>4 010011110=>4 010011111=>4 010100000=>2 010100001=>3 010100010=>3 010100011=>3 010100100=>3 010100101=>4 010100110=>4 010100111=>4 010101000=>3 010101001=>4 010101010=>4 010101011=>4 010101100=>4 010101101=>4 010101110=>4 010101111=>4 010110000=>3 010110001=>4 010110010=>4 010110011=>4 010110100=>4 010110101=>4 010110110=>4 010110111=>4 010111000=>3 010111001=>4 010111010=>4 010111011=>4 010111100=>4 010111101=>4 010111110=>4 010111111=>4 011000000=>2 011000001=>3 011000010=>3 011000011=>3 011000100=>3 011000101=>4 011000110=>4 011000111=>4 011001000=>3 011001001=>4 011001010=>4 011001011=>4 011001100=>4 011001101=>4 011001110=>4 011001111=>4 011010000=>3 011010001=>4 011010010=>4 011010011=>4 011010100=>4 011010101=>4 011010110=>4 011010111=>4 011011000=>3 011011001=>4 011011010=>4 011011011=>4 011011100=>4 011011101=>4 011011110=>4 011011111=>4 011100000=>2 011100001=>3 011100010=>3 011100011=>3 011100100=>3 011100101=>4 011100110=>4 011100111=>4 011101000=>3 011101001=>4 011101010=>4 011101011=>4 011101100=>4 011101101=>4 011101110=>4 011101111=>4 011110000=>3 011110001=>4 011110010=>4 011110011=>4 011110100=>4 011110101=>4 011110110=>4 011110111=>4 011111000=>3 011111001=>4 011111010=>4 011111011=>4 011111100=>4 011111101=>4 011111110=>4 011111111=>4 100000000=>1 100000001=>2 100000010=>2 100000011=>2 100000100=>2 100000101=>3 100000110=>3 100000111=>3 100001000=>2 100001001=>3 100001010=>3 100001011=>3 100001100=>3 100001101=>3 100001110=>3 100001111=>3 100010000=>2 100010001=>3 100010010=>3 100010011=>3 100010100=>3 100010101=>4 100010110=>4 100010111=>4 100011000=>3 100011001=>4 100011010=>4 100011011=>4 100011100=>3 100011101=>4 100011110=>4 100011111=>4 100100000=>2 100100001=>3 100100010=>3 100100011=>3 100100100=>3 100100101=>4 100100110=>4 100100111=>4 100101000=>3 100101001=>4 100101010=>4 100101011=>4 100101100=>4 100101101=>4 100101110=>4 100101111=>4 100110000=>3 100110001=>4 100110010=>4 100110011=>4 100110100=>4 100110101=>4 100110110=>4 100110111=>4 100111000=>3 100111001=>4 100111010=>4 100111011=>4 100111100=>4 100111101=>4 100111110=>4 100111111=>4 101000000=>2 101000001=>3 101000010=>3 101000011=>3 101000100=>3 101000101=>4 101000110=>4 101000111=>4 101001000=>3 101001001=>4 101001010=>4 101001011=>4 101001100=>4 101001101=>4 101001110=>4 101001111=>4 101010000=>3 101010001=>4 101010010=>4 101010011=>4 101010100=>4 101010101=>5 101010110=>5 101010111=>5 101011000=>4 101011001=>5 101011010=>5 101011011=>5 101011100=>4 101011101=>5 101011110=>5 101011111=>5 101100000=>3 101100001=>4 101100010=>4 101100011=>4 101100100=>4 101100101=>5 101100110=>5 101100111=>5 101101000=>4 101101001=>5 101101010=>5 101101011=>5 101101100=>4 101101101=>5 101101110=>5 101101111=>5 101110000=>3 101110001=>4 101110010=>4 101110011=>4 101110100=>4 101110101=>5 101110110=>5 101110111=>5 101111000=>4 101111001=>5 101111010=>5 101111011=>5 101111100=>4 101111101=>5 101111110=>5 101111111=>5 110000000=>2 110000001=>3 110000010=>3 110000011=>3 110000100=>3 110000101=>4 110000110=>4 110000111=>4 110001000=>3 110001001=>4 110001010=>4 110001011=>4 110001100=>4 110001101=>4 110001110=>4 110001111=>4 110010000=>3 110010001=>4 110010010=>4 110010011=>4 110010100=>4 110010101=>5 110010110=>5 110010111=>5 110011000=>4 110011001=>5 110011010=>5 110011011=>5 110011100=>4 110011101=>5 110011110=>5 110011111=>5 110100000=>3 110100001=>4 110100010=>4 110100011=>4 110100100=>4 110100101=>5 110100110=>5 110100111=>5 110101000=>4 110101001=>5 110101010=>5 110101011=>5 110101100=>4 110101101=>5 110101110=>5 110101111=>5 110110000=>3 110110001=>4 110110010=>4 110110011=>4 110110100=>4 110110101=>5 110110110=>5 110110111=>5 110111000=>4 110111001=>5 110111010=>5 110111011=>5 110111100=>4 110111101=>5 110111110=>5 110111111=>5 111000000=>2 111000001=>3 111000010=>3 111000011=>3 111000100=>3 111000101=>4 111000110=>4 111000111=>4 111001000=>3 111001001=>4 111001010=>4 111001011=>4 111001100=>4 111001101=>4 111001110=>4 111001111=>4 111010000=>3 111010001=>4 111010010=>4 111010011=>4 111010100=>4 111010101=>5 111010110=>5 111010111=>5 111011000=>4 111011001=>5 111011010=>5 111011011=>5 111011100=>4 111011101=>5 111011110=>5 111011111=>5 111100000=>3 111100001=>4 111100010=>4 111100011=>4 111100100=>4 111100101=>5 111100110=>5 111100111=>5 111101000=>4 111101001=>5 111101010=>5 111101011=>5 111101100=>4 111101101=>5 111101110=>5 111101111=>5 111110000=>3 111110001=>4 111110010=>4 111110011=>4 111110100=>4 111110101=>5 111110110=>5 111110111=>5 111111000=>4 111111001=>5 111111010=>5 111111011=>5 111111100=>4 111111101=>5 111111110=>5 111111111=>5 0000000001=>1 0010000001=>2 0010101101=>4 0010101011=>4 0010100111=>4 0010011101=>4 0010011011=>4 0010010111=>4 0010001111=>3 0001110101=>4 0001110011=>3 0001101101=>4 0001101011=>4 0001100111=>4 0001011101=>4 0001011011=>4 0001010111=>4 0001001111=>3 0000111101=>3 0000111011=>3 0000110111=>3 0000101111=>3 0000011111=>3 0000000000=>0 0000000110=>2 0000011000=>2 0000011110=>3 0000010010=>2 0001100000=>2 0001100110=>4 0001111000=>3 0001111110=>4 0001110010=>3 0001001000=>2 0001001110=>3 0001000010=>2 0001011010=>4 0010101010=>4 0010101000=>3 0010101100=>4 0010100100=>3 0010010100=>3 0001010100=>3 0010101110=>4 0010100000=>2 0010000100=>2 0000010100=>2 0000000010=>1 0000001110=>2 0000111110=>3 0000000100=>1 0000001010=>2 0000010110=>3 0000101110=>3 0001011110=>4 0001101110=>4 0001110110=>4 0001111010=>4 0001111100=>3 0000000101=>2 0000001101=>2 0000011101=>3 0001111101=>4 0000001001=>2 0000010101=>3 0000101101=>3 0010110001=>4 0001100001=>3 0001000001=>2 0000111111=>3 0001110100=>3 0001101100=>3 0010011100=>3 0001011100=>3 0000111100=>3 0001101010=>4 0010011010=>4 0000111010=>3 0010100110=>4 0010010110=>4 0001010110=>4 0000110110=>3 0010001110=>3 0001101001=>4 0000110001=>3 0000010001=>2 0000001100=>2 0000011001=>3 0000110011=>3 0000110100=>3 0000000011=>1 0000011011=>3 0001101111=>4 0000110101=>3 0010000010=>2 0001000011=>2 0010011110=>4 0001010000=>2 0001000100=>2 0001010101=>4 0001001011=>3 0001100011=>3 0001110111=>4 0010000000=>1 0000000111=>2 0000001111=>2 0001111111=>4 0000001011=>2 0000010111=>3 0001011111=>4 0001111001=>4 0000011010=>3 0000101010=>3 0000100101=>3 0010010001=>3 0010000101=>3 0010010101=>4 0010010011=>3 0000101011=>3 0001000111=>3 0000100111=>3 0010010010=>3 0001001010=>3 0001001001=>3 0001000101=>3 0000100011=>2 0000100100=>2 0000100010=>2 0010110000=>3 0000011100=>2 0000001000=>1 0000101100=>3 0010101111=>4 0000101000=>2 0010001000=>2 0001001100=>3 0001000110=>3 0001100100=>3 0001100010=>3 0000100110=>3 0000110010=>3 0010010000=>2 0001010010=>3 0001000000=>1 0001011001=>4 0001100101=>4 0001110001=>3 0010011001=>4 0010100101=>4 0010101001=>4 0010011111=>4 0000111001=>3 0010001101=>3 0001001101=>3 0001010001=>3 0001010011=>3 0001111011=>4 0010011000=>3 0001011000=>3 0000100000=>1 0010001011=>3 0010000111=>3 0000010011=>2 0010001100=>3 0010000110=>3 0000110000=>2 0000100001=>2 0010100011=>3 0010100001=>3 0000111000=>2 0001101000=>3 0000010000=>1 0000101001=>3 0010100010=>3 0010001001=>3 0010000011=>2 0010001010=>3 0001110000=>2
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Description
A recursive count of subwords of the form 01, 10 and 11.
We iteratively remove the last two bits, and tally the number of times we see 01, 10 or 11. However, if the last two bits are 00, we drop the last 0 and continue. The statistic is 1 for the word 1.
We can recursively define the statistic as
$$ stat(w_1,\dotsc,w_n) = \begin{cases} 0 & \text{if } w \text{ is the empty word}\\ 0 & \text{if } w = (0)\\ 1 & \text{if } w=(1)\\ stat(w_1,\dotsc,w_{n-1}) & \text{if } w_n=w_{n-1}=0\\ 1+stat(w_1,\dotsc,w_{n-2}) & \text{otherwise.} \end{cases} $$
This statistic is Example 4.1 in [1].
References
[1] Shapiro, B., Smirnov, I., Vaintrob, A. Deformed graphical zonotopal algebras arXiv:2204.11331
Code
def statistic(t):
    n = len(t)
    if n == 0:
        return 0
    if n <= 2:
        return max(t)
    if t[-2] == 0 and t[-1] == 0:
        return statistic(t[:-1])

    return 1 + statistic(t[:-2])
Created
Oct 13, 2025 at 15:05 by Per Alexandersson
Updated
Oct 13, 2025 at 18:03 by Martin Rubey