edit this statistic or download as text // json
Identifier
Values
=>
0=>1 1=>1 00=>2 01=>1 10=>1 11=>2 000=>4 001=>2 010=>2 011=>2 100=>2 101=>2 110=>2 111=>4 0000=>8 0001=>4 0010=>3 0011=>4 0100=>3 0101=>3 0110=>3 0111=>4 1000=>4 1001=>3 1010=>3 1011=>3 1100=>4 1101=>3 1110=>4 1111=>8 00000=>16 00001=>8 00010=>6 00011=>8 00100=>6 00101=>5 00110=>5 00111=>8 01000=>6 01001=>4 01010=>5 01011=>5 01100=>5 01101=>4 01110=>5 01111=>8 10000=>8 10001=>5 10010=>4 10011=>5 10100=>5 10101=>5 10110=>4 10111=>6 11000=>8 11001=>5 11010=>5 11011=>6 11100=>8 11101=>6 11110=>8 11111=>16 000000=>32 000001=>16 000010=>12 000011=>16 000100=>11 000101=>10 000110=>10 000111=>16 001000=>11 001001=>8 001010=>8 001011=>8 001100=>10 001101=>7 001110=>9 001111=>16 010000=>12 010001=>7 010010=>7 010011=>7 010100=>8 010101=>8 010110=>6 010111=>10 011000=>10 011001=>6 011010=>6 011011=>8 011100=>9 011101=>7 011110=>9 011111=>16 100000=>16 100001=>9 100010=>7 100011=>9 100100=>8 100101=>6 100110=>6 100111=>10 101000=>10 101001=>6 101010=>8 101011=>8 101100=>7 101101=>7 101110=>7 101111=>12 110000=>16 110001=>9 110010=>7 110011=>10 110100=>8 110101=>8 110110=>8 110111=>11 111000=>16 111001=>10 111010=>10 111011=>11 111100=>16 111101=>12 111110=>16 111111=>32 0000000=>64 0000001=>32 0000010=>24 0000011=>32 0000100=>22 0000101=>20 0000110=>20 0000111=>32 0001000=>22 0001001=>15 0001010=>16 0001011=>16 0001100=>19 0001101=>14 0001110=>18 0001111=>32 0010000=>22 0010001=>13 0010010=>12 0010011=>14 0010100=>14 0010101=>13 0010110=>10 0010111=>16 0011000=>19 0011001=>12 0011010=>11 0011011=>14 0011100=>18 0011101=>13 0011110=>17 0011111=>32 0100000=>24 0100001=>13 0100010=>11 0100011=>13 0100100=>12 0100101=>10 0100110=>9 0100111=>14 0101000=>16 0101001=>10 0101010=>13 0101011=>13 0101100=>11 0101101=>10 0101110=>11 0101111=>20 0110000=>20 0110001=>11 0110010=>9 0110011=>12 0110100=>10 0110101=>10 0110110=>11 0110111=>15 0111000=>18 0111001=>11 0111010=>11 0111011=>13 0111100=>17 0111101=>13 0111110=>17 0111111=>32 1000000=>32 1000001=>17 1000010=>13 1000011=>17 1000100=>13 1000101=>11 1000110=>11 1000111=>18 1001000=>15 1001001=>11 1001010=>10 1001011=>10 1001100=>12 1001101=>9 1001110=>11 1001111=>20 1010000=>20 1010001=>11 1010010=>10 1010011=>11 1010100=>13 1010101=>13 1010110=>10 1010111=>16 1011000=>14 1011001=>9 1011010=>10 1011011=>12 1011100=>13 1011101=>11 1011110=>13 1011111=>24 1100000=>32 1100001=>17 1100010=>13 1100011=>18 1100100=>14 1100101=>11 1100110=>12 1100111=>19 1101000=>16 1101001=>10 1101010=>13 1101011=>14 1101100=>14 1101101=>12 1101110=>13 1101111=>22 1110000=>32 1110001=>18 1110010=>14 1110011=>19 1110100=>16 1110101=>16 1110110=>15 1110111=>22 1111000=>32 1111001=>20 1111010=>20 1111011=>22 1111100=>32 1111101=>24 1111110=>32 1111111=>64 00000000=>128 00000001=>64 00000010=>48 00000011=>64 00000100=>44 00000101=>40 00000110=>40 00000111=>64 00001000=>43 00001001=>30 00001010=>32 00001011=>32 00001100=>38 00001101=>28 00001110=>36 00001111=>64 00010000=>43 00010001=>26 00010010=>23 00010011=>26 00010100=>27 00010101=>26 00010110=>20 00010111=>32 00011000=>38 00011001=>23 00011010=>22 00011011=>28 00011100=>35 00011101=>26 00011110=>34 00011111=>64 00100000=>44 00100001=>24 00100010=>20 00100011=>24 00100100=>23 00100101=>18 00100110=>17 00100111=>28 00101000=>27 00101001=>17 00101010=>21 00101011=>21 00101100=>18 00101101=>17 00101110=>18 00101111=>32 00110000=>38 00110001=>21 00110010=>17 00110011=>24 00110100=>18 00110101=>18 00110110=>19 00110111=>26 00111000=>35 00111001=>22 00111010=>21 00111011=>24 00111100=>34 00111101=>25 00111110=>33 00111111=>64 01000000=>48 01000001=>25 01000010=>20 01000011=>25 01000100=>20 01000101=>17 01000110=>16 01000111=>26 01001000=>23 01001001=>16 01001010=>16 01001011=>17 01001100=>17 01001101=>13 01001110=>16 01001111=>28 01010000=>32 01010001=>18 01010010=>16 01010011=>18 01010100=>21 01010101=>21 01010110=>16 01010111=>26 01011000=>22 01011001=>14 01011010=>16 01011011=>18 01011100=>21 01011101=>17 01011110=>21 01011111=>40 01100000=>40 01100001=>21 01100010=>16 01100011=>22 01100100=>17 01100101=>14 01100110=>16 01100111=>23 01101000=>20 01101001=>13 01101010=>16 01101011=>17 01101100=>19 01101101=>16 01101110=>17 01101111=>30 01110000=>36 01110001=>20 01110010=>16 01110011=>21 01110100=>18 01110101=>18 01110110=>17 01110111=>26 01111000=>34 01111001=>21 01111010=>21 01111011=>24 01111100=>33 01111101=>25 01111110=>33 01111111=>64 10000000=>64 10000001=>33 10000010=>25 10000011=>33 10000100=>24 10000101=>21 10000110=>21 10000111=>34 10001000=>26 10001001=>17 10001010=>18 10001011=>18 10001100=>21 10001101=>16 10001110=>20 10001111=>36 10010000=>30 10010001=>17 10010010=>16 10010011=>19 10010100=>17 10010101=>16 10010110=>13 10010111=>20 10011000=>23 10011001=>16 10011010=>14 10011011=>17 10011100=>22 10011101=>16 10011110=>21 10011111=>40 10100000=>40 10100001=>21 10100010=>17 10100011=>21 10100100=>18 10100101=>16 10100110=>14 10100111=>22 10101000=>26 10101001=>16 10101010=>21 10101011=>21 10101100=>18 10101101=>16 10101110=>18 10101111=>32 10110000=>28 10110001=>16 10110010=>13 10110011=>17 10110100=>17 10110101=>16 10110110=>16 10110111=>23 10111000=>26 10111001=>16 10111010=>17 10111011=>20 10111100=>25 10111101=>20 10111110=>25 10111111=>48 11000000=>64 11000001=>33 11000010=>25 11000011=>34 11000100=>24 11000101=>21 11000110=>22 11000111=>35 11001000=>26 11001001=>19 11001010=>18 11001011=>18 11001100=>24 11001101=>17 11001110=>21 11001111=>38 11010000=>32 11010001=>18 11010010=>17 11010011=>18 11010100=>21 11010101=>21 11010110=>17 11010111=>27 11011000=>28 11011001=>17 11011010=>18 11011011=>23 11011100=>24 11011101=>20 11011110=>24 11011111=>44 11100000=>64 11100001=>34 11100010=>26 11100011=>35 11100100=>28 11100101=>22 11100110=>23 11100111=>38 11101000=>32 11101001=>20 11101010=>26 11101011=>27 11101100=>26 11101101=>23 11101110=>26 11101111=>43 11110000=>64 11110001=>36 11110010=>28 11110011=>38 11110100=>32 11110101=>32 11110110=>30 11110111=>43 11111000=>64 11111001=>40 11111010=>40 11111011=>44 11111100=>64 11111101=>48 11111110=>64 11111111=>128 000000000=>256 000000001=>128 000000010=>96 000000011=>128 000000100=>88 000000101=>80 000000110=>80 000000111=>128 000001000=>86 000001001=>60 000001010=>64 000001011=>64 000001100=>76 000001101=>56 000001110=>72 000001111=>128 000010000=>86 000010001=>51 000010010=>46 000010011=>52 000010100=>54 000010101=>52 000010110=>40 000010111=>64 000011000=>75 000011001=>46 000011010=>44 000011011=>56 000011100=>70 000011101=>52 000011110=>68 000011111=>128 000100000=>86 000100001=>47 000100010=>39 000100011=>48 000100100=>43 000100101=>34 000100110=>32 000100111=>52 000101000=>54 000101001=>33 000101010=>42 000101011=>42 000101100=>36 000101101=>34 000101110=>36 000101111=>64 000110000=>75 000110001=>42 000110010=>33 000110011=>46 000110100=>36 000110101=>36 000110110=>38 000110111=>52 000111000=>70 000111001=>43 000111010=>42 000111011=>48 000111100=>67 000111101=>50 000111110=>66 000111111=>128 001000000=>88 001000001=>46 001000010=>36 001000011=>46 001000100=>37 001000101=>31 001000110=>30 001000111=>48 001001000=>43 001001001=>31 001001010=>29 001001011=>30 001001100=>33 001001101=>25 001001110=>31 001001111=>56 001010000=>54 001010001=>30 001010010=>27 001010011=>31 001010100=>35 001010101=>34 001010110=>26 001010111=>42 001011000=>36 001011001=>23 001011010=>26 001011011=>30 001011100=>34 001011101=>28 001011110=>34 001011111=>64 001100000=>76 001100001=>40 001100010=>31 001100011=>42 001100100=>33 001100101=>27 001100110=>30 001100111=>46 001101000=>36 001101001=>23 001101010=>29 001101011=>31 001101100=>34 001101101=>28 001101110=>30 001101111=>52 001110000=>70 001110001=>39 001110010=>31 001110011=>42 001110100=>34 001110101=>34 001110110=>32 001110111=>48 001111000=>67 001111001=>42 001111010=>41 001111011=>46 001111100=>66 001111101=>49 001111110=>65 001111111=>128 010000000=>96 010000001=>49 010000010=>38 010000011=>49 010000100=>36 010000101=>32 010000110=>31 010000111=>50 010001000=>39 010001001=>26 010001010=>27 010001011=>28 010001100=>31 010001101=>23 010001110=>29 010001111=>52 010010000=>46 010010001=>26 010010010=>26 010010011=>28 010010100=>27 010010101=>26 010010110=>21 010010111=>34 010011000=>33 010011001=>22 010011010=>20 010011011=>25 010011100=>31 010011101=>23 010011110=>30 010011111=>56 010100000=>64 010100001=>34 010100010=>27 010100011=>34 010100100=>29 010100101=>25 010100110=>23 010100111=>36 010101000=>42 010101001=>26 010101010=>34 010101011=>34 010101100=>29 010101101=>26 010101110=>29 010101111=>52 010110000=>44 010110001=>25 010110010=>20 010110011=>27 010110100=>26 010110101=>25 010110110=>24 010110111=>34 010111000=>42 010111001=>26 010111010=>28 010111011=>31 010111100=>41 010111101=>32 010111110=>41 010111111=>80 011000000=>80 011000001=>41 011000010=>31 011000011=>42 011000100=>30 011000101=>26 011000110=>28 011000111=>43 011001000=>32 011001001=>23 011001010=>23 011001011=>23 011001100=>30 011001101=>22 011001110=>26 011001111=>46 011010000=>40 011010001=>23 011010010=>21 011010011=>23 011010100=>26 011010101=>26 011010110=>22 011010111=>33 011011000=>38 011011001=>23 011011010=>24 011011011=>31 011011100=>32 011011101=>26 011011110=>32 011011111=>60 011100000=>72 011100001=>38 011100010=>29 011100011=>39 011100100=>31 011100101=>25 011100110=>26 011100111=>42 011101000=>36 011101001=>23 011101010=>29 011101011=>30 011101100=>30 011101101=>26 011101110=>31 011101111=>51 011110000=>68 011110001=>38 011110010=>30 011110011=>40 011110100=>34 011110101=>34 011110110=>32 011110111=>47 011111000=>66 011111001=>41 011111010=>41 011111011=>46 011111100=>65 011111101=>49 011111110=>65 011111111=>128 100000000=>128 100000001=>65 100000010=>49 100000011=>65 100000100=>46 100000101=>41 100000110=>41 100000111=>66 100001000=>47 100001001=>32 100001010=>34 100001011=>34 100001100=>40 100001101=>30 100001110=>38 100001111=>68 100010000=>51 100010001=>31 100010010=>26 100010011=>30 100010100=>30 100010101=>29 100010110=>23 100010111=>36 100011000=>42 100011001=>26 100011010=>25 100011011=>31 100011100=>39 100011101=>29 100011110=>38 100011111=>72 100100000=>60 100100001=>32 100100010=>26 100100011=>32 100100100=>31 100100101=>24 100100110=>23 100100111=>38 100101000=>33 100101001=>22 100101010=>26 100101011=>26 100101100=>23 100101101=>21 100101110=>23 100101111=>40 100110000=>46 100110001=>26 100110010=>22 100110011=>30 100110100=>23 100110101=>23 100110110=>23 100110111=>32 100111000=>43 100111001=>28 100111010=>26 100111011=>30 100111100=>42 100111101=>31 100111110=>41 100111111=>80 101000000=>80 101000001=>41 101000010=>32 101000011=>41 101000100=>31 101000101=>28 101000110=>26 101000111=>42 101001000=>34 101001001=>24 101001010=>25 101001011=>26 101001100=>27 101001101=>20 101001110=>25 101001111=>44 101010000=>52 101010001=>29 101010010=>26 101010011=>29 101010100=>34 101010101=>34 101010110=>26 101010111=>42 101011000=>36 101011001=>23 101011010=>25 101011011=>29 101011100=>34 101011101=>27 101011110=>34 101011111=>64 101100000=>56 101100001=>30 101100010=>23 101100011=>31 101100100=>25 101100101=>20 101100110=>22 101100111=>33 101101000=>34 101101001=>21 101101010=>26 101101011=>27 101101100=>28 101101101=>26 101101110=>26 101101111=>46 101110000=>52 101110001=>29 101110010=>23 101110011=>31 101110100=>28 101110101=>27 101110110=>26 101110111=>39 101111000=>50 101111001=>31 101111010=>32 101111011=>36 101111100=>49 101111101=>38 101111110=>49 101111111=>96 110000000=>128 110000001=>65 110000010=>49 110000011=>66 110000100=>46 110000101=>41 110000110=>42 110000111=>67 110001000=>48 110001001=>32 110001010=>34 110001011=>34 110001100=>42 110001101=>31 110001110=>39 110001111=>70 110010000=>52 110010001=>30 110010010=>28 110010011=>34 110010100=>31 110010101=>29 110010110=>23 110010111=>36 110011000=>46 110011001=>30 110011010=>27 110011011=>33 110011100=>42 110011101=>31 110011110=>40 110011111=>76 110100000=>64 110100001=>34 110100010=>28 110100011=>34 110100100=>30 110100101=>26 110100110=>23 110100111=>36 110101000=>42 110101001=>26 110101010=>34 110101011=>35 110101100=>31 110101101=>27 110101110=>30 110101111=>54 110110000=>56 110110001=>31 110110010=>25 110110011=>33 110110100=>30 110110101=>29 110110110=>31 110110111=>43 110111000=>48 110111001=>30 110111010=>31 110111011=>37 110111100=>46 110111101=>36 110111110=>46 110111111=>88 111000000=>128 111000001=>66 111000010=>50 111000011=>67 111000100=>48 111000101=>42 111000110=>43 111000111=>70 111001000=>52 111001001=>38 111001010=>36 111001011=>36 111001100=>46 111001101=>33 111001110=>42 111001111=>75 111010000=>64 111010001=>36 111010010=>34 111010011=>36 111010100=>42 111010101=>42 111010110=>33 111010111=>54 111011000=>52 111011001=>32 111011010=>34 111011011=>43 111011100=>48 111011101=>39 111011110=>47 111011111=>86 111100000=>128 111100001=>68 111100010=>52 111100011=>70 111100100=>56 111100101=>44 111100110=>46 111100111=>75 111101000=>64 111101001=>40 111101010=>52 111101011=>54 111101100=>52 111101101=>46 111101110=>51 111101111=>86 111110000=>128 111110001=>72 111110010=>56 111110011=>76 111110100=>64 111110101=>64 111110110=>60 111110111=>86 111111000=>128 111111001=>80 111111010=>80 111111011=>88 111111100=>128 111111101=>96 111111110=>128 111111111=>256 0000000001=>256 0010000001=>90 0010101101=>42 0010101011=>55 0010100111=>62 0010011101=>45 0010011011=>48 0010010111=>60 0010001111=>96 0001110101=>68 0001110011=>82 0001101101=>56 0001101011=>62 0001100111=>88 0001011101=>56 0001011011=>60 0001010111=>84 0001001111=>104 0000111101=>100 0000111011=>96 0000110111=>104 0000101111=>128 0000011111=>256 0000000000=>512 0000000110=>160 0000011000=>150 0000011110=>136 0000010010=>92 0001100000=>150 0001100110=>58 0001111000=>134 0001111110=>130 0001110010=>61 0001001000=>82 0001001110=>58 0001000010=>71 0001011010=>52 0010101010=>55 0010101000=>69 0010101100=>47 0010100100=>49 0010010100=>49 0001010100=>69 0010101110=>47 0010100000=>108 0010000100=>67 0000010100=>108 0000000010=>192 0000001110=>144 0000111110=>132 0000000100=>176 0000001010=>128 0000010110=>80 0000101110=>72 0001011110=>68 0001101110=>60 0001110110=>64 0001111010=>82 0001111100=>131 0000000101=>160 0000001101=>112 0000011101=>104 0001111101=>98 0000001001=>120 0000010101=>104 0000101101=>68 0010110001=>41 0001100001=>79 0001000001=>90 0000111111=>256 0001110100=>68 0001101100=>67 0010011100=>61 0001011100=>68 0000111100=>134 0001101010=>58 0010011010=>39 0000111010=>84 0010100110=>39 0010010110=>38 0001010110=>52 0000110110=>76 0010001110=>54 0001101001=>46 0000110001=>83 0000010001=>102 0000001100=>152 0000011001=>92 0000110011=>92 0000110100=>72 0000000011=>256 0000011011=>112 0001101111=>104 0000110101=>72 0010000010=>69 0001000011=>90 0010011110=>59 0001010000=>107 0001000100=>72 0001010101=>68 0001001011=>57 0001100011=>84 0001110111=>96 0010000000=>176 0000000111=>256 0000001111=>256 0001111111=>256 0000001011=>128 0000010111=>128 0001011111=>128 0001111001=>83 0000011010=>88 0000101010=>84 0000100101=>68 0010010001=>49 0010000101=>58 0010010101=>47 0010010011=>54 0000101011=>84 0001000111=>96 0000100111=>104 0010010010=>47 0001001010=>55 0001001001=>58 0001000101=>61 0000100011=>94 0000100100=>86 0000100010=>77 0010110000=>72 0000011100=>140 0000001000=>172 0000101100=>72 0010101111=>84 0000101000=>107 0010001000=>72 0001001100=>62 0001000110=>59 0001100100=>64 0001100010=>61 0000100110=>64 0000110010=>66 0010010000=>86 0001010010=>53 0001000000=>172 0001011001=>46 0001100101=>52 0001110001=>78 0010011001=>43 0010100101=>43 0010101001=>43 0010011111=>112 0000111001=>86 0010001101=>43 0001001101=>47 0001010001=>60 0001010011=>60 0001111011=>92 0010011000=>64 0001011000=>72 0000100000=>171 0010001011=>51 0010000111=>92 0000010011=>104 0010001100=>57 0010000110=>57 0000110000=>150 0000100001=>94 0010100011=>57 0010100001=>57 0000111000=>139 0001101000=>72 0000010000=>171 0000101001=>66 0010100010=>46 0010001001=>48 0010000011=>90 0010001010=>50 0001110000=>139
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of distinct palindromic decompositions 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.
References
[1] mo201205
Code
def statistic(w):
    return len(paldecs(w))

def paldecs(w):
    """
    Return all distinct palindromic decompositions of w.

    sage: w = Word([1,1])
    sage: paldecs(w)
    [[word: 1, word: 1], [word: 11]]

    sage: w = Word([1,0,1])
    sage: paldecs(w)
    [[word: 1, word: 0, word: 1], [word: 101]]

    sage: w = Word([1,0,1,0,0,1])
    sage: paldecs(w)
    [[word: 1, word: 0, word: 1, word: 0, word: 0, word: 1],
     [word: 1, word: 0, word: 1, word: 00, word: 1],
     [word: 1, word: 0, word: 1001],
     [word: 1, word: 010, word: 0, word: 1],
     [word: 101, word: 0, word: 0, word: 1],
     [word: 101, word: 00, word: 1]]

    sage: w = Word("referee")
    sage: paldecs(w)
    [[word: r, word: e, word: f, word: e, word: r, word: e, word: e],
     [word: r, word: e, word: f, word: e, word: r, word: ee],
     [word: r, word: e, word: f, word: ere, word: e],
     [word: r, word: efe, word: r, word: e, word: e],
     [word: r, word: efe, word: r, word: ee],
     [word: refer, word: e, word: e],
     [word: refer, word: ee]]

    """
    if len(w) == 0:
        return [[]]
    P1 = w.palindrome_prefixes()
    result = []
    for p1 in P1:
        l = len(p1)
        if l > 0:
            P2 = paldecs(w[l:])
            for p2 in P2:
                result.append([p1] + p2)
    return result
Created
Oct 16, 2016 at 21:12 by Martin Rubey
Updated
Oct 16, 2016 at 21:12 by Martin Rubey