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-----------------------------------------------------------------------------
Statistic identifier: St000630
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The length of the shortest palindromic decomposition 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] მáƒáƒ›áƒ£áƒ™áƒ ჯიბლáƒáƒ«áƒ” Combinatorics of palindromic decompositions [[MathOverflow:201205]]
[2] Ravsky, A. On the palindromic decomposition of binary words [[arXiv:1004.1278]]
-----------------------------------------------------------------------------
Code:
def statistic(w):
return min(len(p) for p in 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
-----------------------------------------------------------------------------
Statistic values:
0 => 1
1 => 1
00 => 1
01 => 2
10 => 2
11 => 1
000 => 1
001 => 2
010 => 1
011 => 2
100 => 2
101 => 1
110 => 2
111 => 1
0000 => 1
0001 => 2
0010 => 2
0011 => 2
0100 => 2
0101 => 2
0110 => 1
0111 => 2
1000 => 2
1001 => 1
1010 => 2
1011 => 2
1100 => 2
1101 => 2
1110 => 2
1111 => 1
00000 => 1
00001 => 2
00010 => 2
00011 => 2
00100 => 1
00101 => 2
00110 => 2
00111 => 2
01000 => 2
01001 => 2
01010 => 1
01011 => 2
01100 => 2
01101 => 2
01110 => 1
01111 => 2
10000 => 2
10001 => 1
10010 => 2
10011 => 2
10100 => 2
10101 => 1
10110 => 2
10111 => 2
11000 => 2
11001 => 2
11010 => 2
11011 => 1
11100 => 2
11101 => 2
11110 => 2
11111 => 1
000000 => 1
000001 => 2
000010 => 2
000011 => 2
000100 => 2
000101 => 2
000110 => 2
000111 => 2
001000 => 2
001001 => 2
001010 => 2
001011 => 3
001100 => 1
001101 => 3
001110 => 2
001111 => 2
010000 => 2
010001 => 2
010010 => 1
010011 => 3
010100 => 2
010101 => 2
010110 => 3
010111 => 2
011000 => 2
011001 => 3
011010 => 3
011011 => 2
011100 => 2
011101 => 2
011110 => 1
011111 => 2
100000 => 2
100001 => 1
100010 => 2
100011 => 2
100100 => 2
100101 => 3
100110 => 3
100111 => 2
101000 => 2
101001 => 3
101010 => 2
101011 => 2
101100 => 3
101101 => 1
101110 => 2
101111 => 2
110000 => 2
110001 => 2
110010 => 3
110011 => 1
110100 => 3
110101 => 2
110110 => 2
110111 => 2
111000 => 2
111001 => 2
111010 => 2
111011 => 2
111100 => 2
111101 => 2
111110 => 2
111111 => 1
0000000 => 1
0000001 => 2
0000010 => 2
0000011 => 2
0000100 => 2
0000101 => 2
0000110 => 2
0000111 => 2
0001000 => 1
0001001 => 2
0001010 => 2
0001011 => 3
0001100 => 2
0001101 => 3
0001110 => 2
0001111 => 2
0010000 => 2
0010001 => 2
0010010 => 2
0010011 => 2
0010100 => 1
0010101 => 2
0010110 => 3
0010111 => 3
0011000 => 2
0011001 => 2
0011010 => 3
0011011 => 2
0011100 => 1
0011101 => 3
0011110 => 2
0011111 => 2
0100000 => 2
0100001 => 2
0100010 => 1
0100011 => 3
0100100 => 2
0100101 => 2
0100110 => 2
0100111 => 3
0101000 => 2
0101001 => 2
0101010 => 1
0101011 => 2
0101100 => 3
0101101 => 2
0101110 => 3
0101111 => 2
0110000 => 2
0110001 => 3
0110010 => 2
0110011 => 2
0110100 => 3
0110101 => 2
0110110 => 1
0110111 => 2
0111000 => 2
0111001 => 3
0111010 => 3
0111011 => 2
0111100 => 2
0111101 => 2
0111110 => 1
0111111 => 2
1000000 => 2
1000001 => 1
1000010 => 2
1000011 => 2
1000100 => 2
1000101 => 3
1000110 => 3
1000111 => 2
1001000 => 2
1001001 => 1
1001010 => 2
1001011 => 3
1001100 => 2
1001101 => 2
1001110 => 3
1001111 => 2
1010000 => 2
1010001 => 3
1010010 => 2
1010011 => 3
1010100 => 2
1010101 => 1
1010110 => 2
1010111 => 2
1011000 => 3
1011001 => 2
1011010 => 2
1011011 => 2
1011100 => 3
1011101 => 1
1011110 => 2
1011111 => 2
1100000 => 2
1100001 => 2
1100010 => 3
1100011 => 1
1100100 => 2
1100101 => 3
1100110 => 2
1100111 => 2
1101000 => 3
1101001 => 3
1101010 => 2
1101011 => 1
1101100 => 2
1101101 => 2
1101110 => 2
1101111 => 2
1110000 => 2
1110001 => 2
1110010 => 3
1110011 => 2
1110100 => 3
1110101 => 2
1110110 => 2
1110111 => 1
1111000 => 2
1111001 => 2
1111010 => 2
1111011 => 2
1111100 => 2
1111101 => 2
1111110 => 2
1111111 => 1
00000000 => 1
00000001 => 2
00000010 => 2
00000011 => 2
00000100 => 2
00000101 => 2
00000110 => 2
00000111 => 2
00001000 => 2
00001001 => 2
00001010 => 2
00001011 => 3
00001100 => 2
00001101 => 3
00001110 => 2
00001111 => 2
00010000 => 2
00010001 => 2
00010010 => 2
00010011 => 3
00010100 => 2
00010101 => 2
00010110 => 3
00010111 => 3
00011000 => 1
00011001 => 3
00011010 => 3
00011011 => 2
00011100 => 2
00011101 => 3
00011110 => 2
00011111 => 2
00100000 => 2
00100001 => 2
00100010 => 2
00100011 => 3
00100100 => 1
00100101 => 2
00100110 => 3
00100111 => 2
00101000 => 2
00101001 => 2
00101010 => 2
00101011 => 3
00101100 => 4
00101101 => 2
00101110 => 3
00101111 => 3
00110000 => 2
00110001 => 3
00110010 => 3
00110011 => 2
00110100 => 4
00110101 => 3
00110110 => 2
00110111 => 3
00111000 => 2
00111001 => 2
00111010 => 3
00111011 => 3
00111100 => 1
00111101 => 3
00111110 => 2
00111111 => 2
01000000 => 2
01000001 => 2
01000010 => 1
01000011 => 3
01000100 => 2
01000101 => 2
01000110 => 3
01000111 => 3
01001000 => 2
01001001 => 2
01001010 => 2
01001011 => 2
01001100 => 3
01001101 => 3
01001110 => 2
01001111 => 3
01010000 => 2
01010001 => 2
01010010 => 2
01010011 => 3
01010100 => 2
01010101 => 2
01010110 => 3
01010111 => 2
01011000 => 3
01011001 => 3
01011010 => 1
01011011 => 2
01011100 => 3
01011101 => 2
01011110 => 3
01011111 => 2
01100000 => 2
01100001 => 3
01100010 => 3
01100011 => 2
01100100 => 3
01100101 => 3
01100110 => 1
01100111 => 3
01101000 => 3
01101001 => 2
01101010 => 3
01101011 => 2
01101100 => 2
01101101 => 2
01101110 => 3
01101111 => 2
01110000 => 2
01110001 => 3
01110010 => 2
01110011 => 3
01110100 => 3
01110101 => 2
01110110 => 3
01110111 => 2
01111000 => 2
01111001 => 3
01111010 => 3
01111011 => 2
01111100 => 2
01111101 => 2
01111110 => 1
01111111 => 2
10000000 => 2
10000001 => 1
10000010 => 2
10000011 => 2
10000100 => 2
10000101 => 3
10000110 => 3
10000111 => 2
10001000 => 2
10001001 => 3
10001010 => 2
10001011 => 3
10001100 => 3
10001101 => 2
10001110 => 3
10001111 => 2
10010000 => 2
10010001 => 3
10010010 => 2
10010011 => 2
10010100 => 2
10010101 => 3
10010110 => 2
10010111 => 3
10011000 => 3
10011001 => 1
10011010 => 3
10011011 => 3
10011100 => 2
10011101 => 3
10011110 => 3
10011111 => 2
10100000 => 2
10100001 => 3
10100010 => 2
10100011 => 3
10100100 => 2
10100101 => 1
10100110 => 3
10100111 => 3
10101000 => 2
10101001 => 3
10101010 => 2
10101011 => 2
10101100 => 3
10101101 => 2
10101110 => 2
10101111 => 2
10110000 => 3
10110001 => 2
10110010 => 3
10110011 => 3
10110100 => 2
10110101 => 2
10110110 => 2
10110111 => 2
10111000 => 3
10111001 => 3
10111010 => 2
10111011 => 2
10111100 => 3
10111101 => 1
10111110 => 2
10111111 => 2
11000000 => 2
11000001 => 2
11000010 => 3
11000011 => 1
11000100 => 3
11000101 => 3
11000110 => 2
11000111 => 2
11001000 => 3
11001001 => 2
11001010 => 3
11001011 => 4
11001100 => 2
11001101 => 3
11001110 => 3
11001111 => 2
11010000 => 3
11010001 => 3
11010010 => 2
11010011 => 4
11010100 => 3
11010101 => 2
11010110 => 2
11010111 => 2
11011000 => 2
11011001 => 3
11011010 => 2
11011011 => 1
11011100 => 3
11011101 => 2
11011110 => 2
11011111 => 2
11100000 => 2
11100001 => 2
11100010 => 3
11100011 => 2
11100100 => 2
11100101 => 3
11100110 => 3
11100111 => 1
11101000 => 3
11101001 => 3
11101010 => 2
11101011 => 2
11101100 => 3
11101101 => 2
11101110 => 2
11101111 => 2
11110000 => 2
11110001 => 2
11110010 => 3
11110011 => 2
11110100 => 3
11110101 => 2
11110110 => 2
11110111 => 2
11111000 => 2
11111001 => 2
11111010 => 2
11111011 => 2
11111100 => 2
11111101 => 2
11111110 => 2
11111111 => 1
000000000 => 1
000000001 => 2
000000010 => 2
000000011 => 2
000000100 => 2
000000101 => 2
000000110 => 2
000000111 => 2
000001000 => 2
000001001 => 2
000001010 => 2
000001011 => 3
000001100 => 2
000001101 => 3
000001110 => 2
000001111 => 2
000010000 => 1
000010001 => 2
000010010 => 2
000010011 => 3
000010100 => 2
000010101 => 2
000010110 => 3
000010111 => 3
000011000 => 2
000011001 => 3
000011010 => 3
000011011 => 2
000011100 => 2
000011101 => 3
000011110 => 2
000011111 => 2
000100000 => 2
000100001 => 2
000100010 => 2
000100011 => 2
000100100 => 2
000100101 => 3
000100110 => 3
000100111 => 3
000101000 => 1
000101001 => 3
000101010 => 2
000101011 => 3
000101100 => 4
000101101 => 2
000101110 => 3
000101111 => 3
000110000 => 2
000110001 => 2
000110010 => 3
000110011 => 2
000110100 => 4
000110101 => 3
000110110 => 2
000110111 => 3
000111000 => 1
000111001 => 3
000111010 => 3
000111011 => 3
000111100 => 2
000111101 => 3
000111110 => 2
000111111 => 2
001000000 => 2
001000001 => 2
001000010 => 2
001000011 => 3
001000100 => 1
001000101 => 3
001000110 => 2
001000111 => 3
001001000 => 2
001001001 => 2
001001010 => 3
001001011 => 3
001001100 => 3
001001101 => 3
001001110 => 3
001001111 => 2
001010000 => 2
001010001 => 3
001010010 => 3
001010011 => 2
001010100 => 1
001010101 => 2
001010110 => 3
001010111 => 3
001011000 => 4
001011001 => 3
001011010 => 2
001011011 => 3
001011100 => 4
001011101 => 2
001011110 => 3
001011111 => 3
001100000 => 2
001100001 => 3
001100010 => 2
001100011 => 2
001100100 => 3
001100101 => 2
001100110 => 2
001100111 => 2
001101000 => 4
001101001 => 3
001101010 => 3
001101011 => 2
001101100 => 1
001101101 => 3
001101110 => 3
001101111 => 3
001110000 => 2
001110001 => 3
001110010 => 3
001110011 => 2
001110100 => 4
001110101 => 3
001110110 => 3
001110111 => 2
001111000 => 2
001111001 => 2
001111010 => 3
001111011 => 3
001111100 => 1
001111101 => 3
001111110 => 2
001111111 => 2
010000000 => 2
010000001 => 2
010000010 => 1
010000011 => 3
010000100 => 2
010000101 => 2
010000110 => 3
010000111 => 3
010001000 => 2
010001001 => 3
010001010 => 3
010001011 => 2
010001100 => 2
010001101 => 3
010001110 => 3
010001111 => 3
010010000 => 2
010010001 => 3
010010010 => 1
010010011 => 3
010010100 => 3
010010101 => 2
010010110 => 3
010010111 => 2
010011000 => 3
010011001 => 2
010011010 => 4
010011011 => 3
010011100 => 3
010011101 => 3
010011110 => 2
010011111 => 3
010100000 => 2
010100001 => 2
010100010 => 3
010100011 => 3
010100100 => 3
010100101 => 2
010100110 => 2
010100111 => 3
010101000 => 2
010101001 => 2
010101010 => 1
010101011 => 2
010101100 => 3
010101101 => 2
010101110 => 3
010101111 => 2
010110000 => 3
010110001 => 3
010110010 => 4
010110011 => 2
010110100 => 2
010110101 => 2
010110110 => 3
010110111 => 3
010111000 => 3
010111001 => 3
010111010 => 1
010111011 => 3
010111100 => 3
010111101 => 2
010111110 => 3
010111111 => 2
011000000 => 2
011000001 => 3
011000010 => 3
011000011 => 2
011000100 => 2
011000101 => 3
011000110 => 1
011000111 => 3
011001000 => 3
011001001 => 3
011001010 => 2
011001011 => 3
011001100 => 2
011001101 => 2
011001110 => 2
011001111 => 3
011010000 => 3
011010001 => 2
011010010 => 3
011010011 => 3
011010100 => 3
011010101 => 2
011010110 => 1
011010111 => 3
011011000 => 2
011011001 => 3
011011010 => 3
011011011 => 2
011011100 => 3
011011101 => 3
011011110 => 3
011011111 => 2
011100000 => 2
011100001 => 3
011100010 => 3
011100011 => 3
011100100 => 3
011100101 => 3
011100110 => 2
011100111 => 2
011101000 => 3
011101001 => 2
011101010 => 3
011101011 => 3
011101100 => 3
011101101 => 3
011101110 => 1
011101111 => 2
011110000 => 2
011110001 => 3
011110010 => 2
011110011 => 3
011110100 => 3
011110101 => 2
011110110 => 3
011110111 => 2
011111000 => 2
011111001 => 3
011111010 => 3
011111011 => 2
011111100 => 2
011111101 => 2
011111110 => 1
011111111 => 2
100000000 => 2
100000001 => 1
100000010 => 2
100000011 => 2
100000100 => 2
100000101 => 3
100000110 => 3
100000111 => 2
100001000 => 2
100001001 => 3
100001010 => 2
100001011 => 3
100001100 => 3
100001101 => 2
100001110 => 3
100001111 => 2
100010000 => 2
100010001 => 1
100010010 => 3
100010011 => 3
100010100 => 3
100010101 => 3
100010110 => 2
100010111 => 3
100011000 => 2
100011001 => 2
100011010 => 3
100011011 => 3
100011100 => 3
100011101 => 3
100011110 => 3
100011111 => 2
100100000 => 2
100100001 => 3
100100010 => 3
100100011 => 3
100100100 => 2
100100101 => 3
100100110 => 3
100100111 => 2
100101000 => 3
100101001 => 1
100101010 => 2
100101011 => 3
100101100 => 3
100101101 => 3
100101110 => 2
100101111 => 3
100110000 => 3
100110001 => 2
100110010 => 2
100110011 => 2
100110100 => 3
100110101 => 2
100110110 => 3
100110111 => 3
100111000 => 3
100111001 => 1
100111010 => 3
100111011 => 2
100111100 => 2
100111101 => 3
100111110 => 3
100111111 => 2
101000000 => 2
101000001 => 3
101000010 => 2
101000011 => 3
101000100 => 3
101000101 => 1
101000110 => 3
101000111 => 3
101001000 => 3
101001001 => 3
101001010 => 2
101001011 => 2
101001100 => 2
101001101 => 4
101001110 => 3
101001111 => 3
101010000 => 2
101010001 => 3
101010010 => 2
101010011 => 3
101010100 => 2
101010101 => 1
101010110 => 2
101010111 => 2
101011000 => 3
101011001 => 2
101011010 => 2
101011011 => 3
101011100 => 3
101011101 => 3
101011110 => 2
101011111 => 2
101100000 => 3
101100001 => 2
101100010 => 3
101100011 => 3
101100100 => 3
101100101 => 4
101100110 => 2
101100111 => 3
101101000 => 2
101101001 => 3
101101010 => 2
101101011 => 3
101101100 => 3
101101101 => 1
101101110 => 3
101101111 => 2
101110000 => 3
101110001 => 3
101110010 => 3
101110011 => 2
101110100 => 2
101110101 => 3
101110110 => 3
101110111 => 2
101111000 => 3
101111001 => 3
101111010 => 2
101111011 => 2
101111100 => 3
101111101 => 1
101111110 => 2
101111111 => 2
110000000 => 2
110000001 => 2
110000010 => 3
110000011 => 1
110000100 => 3
110000101 => 3
110000110 => 2
110000111 => 2
110001000 => 2
110001001 => 3
110001010 => 3
110001011 => 4
110001100 => 2
110001101 => 3
110001110 => 3
110001111 => 2
110010000 => 3
110010001 => 3
110010010 => 3
110010011 => 1
110010100 => 2
110010101 => 3
110010110 => 3
110010111 => 4
110011000 => 2
110011001 => 2
110011010 => 2
110011011 => 3
110011100 => 2
110011101 => 2
110011110 => 3
110011111 => 2
110100000 => 3
110100001 => 3
110100010 => 2
110100011 => 4
110100100 => 3
110100101 => 2
110100110 => 3
110100111 => 4
110101000 => 3
110101001 => 3
110101010 => 2
110101011 => 1
110101100 => 2
110101101 => 3
110101110 => 3
110101111 => 2
110110000 => 2
110110001 => 3
110110010 => 3
110110011 => 3
110110100 => 3
110110101 => 3
110110110 => 2
110110111 => 2
110111000 => 3
110111001 => 2
110111010 => 3
110111011 => 1
110111100 => 3
110111101 => 2
110111110 => 2
110111111 => 2
111000000 => 2
111000001 => 2
111000010 => 3
111000011 => 2
111000100 => 3
111000101 => 3
111000110 => 3
111000111 => 1
111001000 => 3
111001001 => 2
111001010 => 3
111001011 => 4
111001100 => 2
111001101 => 3
111001110 => 2
111001111 => 2
111010000 => 3
111010001 => 3
111010010 => 2
111010011 => 4
111010100 => 3
111010101 => 2
111010110 => 3
111010111 => 1
111011000 => 3
111011001 => 3
111011010 => 3
111011011 => 2
111011100 => 2
111011101 => 2
111011110 => 2
111011111 => 2
111100000 => 2
111100001 => 2
111100010 => 3
111100011 => 2
111100100 => 2
111100101 => 3
111100110 => 3
111100111 => 2
111101000 => 3
111101001 => 3
111101010 => 2
111101011 => 2
111101100 => 3
111101101 => 2
111101110 => 2
111101111 => 1
111110000 => 2
111110001 => 2
111110010 => 3
111110011 => 2
111110100 => 3
111110101 => 2
111110110 => 2
111110111 => 2
111111000 => 2
111111001 => 2
111111010 => 2
111111011 => 2
111111100 => 2
111111101 => 2
111111110 => 2
111111111 => 1
1010101010 => 2
1010101100 => 3
1010110010 => 3
1010110100 => 3
1010111000 => 3
1011001010 => 3
1011001100 => 3
1011010010 => 3
1011010100 => 3
1011011000 => 3
1011100010 => 4
1011100100 => 3
1011101000 => 2
1011110000 => 3
1100101010 => 3
1100101100 => 4
1100110010 => 3
1100110100 => 3
1100111000 => 3
1101001010 => 3
1101001100 => 3
1101010010 => 3
1101010100 => 3
1101011000 => 2
1101100010 => 3
1101100100 => 2
1101101000 => 3
1101110000 => 3
1110001010 => 3
1110001100 => 3
1110010010 => 3
1110010100 => 2
1110011000 => 3
1110100010 => 2
1110100100 => 3
1110101000 => 3
1110110000 => 3
1111000010 => 3
1111000100 => 3
1111001000 => 3
1111010000 => 3
1111100000 => 2
1111111110 => 2
1111111101 => 2
1111000001 => 2
1110111110 => 2
1110100001 => 3
1110011110 => 3
1110010001 => 3
1110001001 => 3
1110000101 => 3
1101111110 => 2
1101100001 => 3
1101010001 => 3
1101001110 => 3
1101001001 => 3
1101000101 => 2
1100110001 => 3
1100101001 => 2
1100100110 => 2
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-----------------------------------------------------------------------------
Created: Oct 16, 2016 at 21:19 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Dec 30, 2017 at 22:39 by Martin Rubey