Identifier
- St000631: Binary words ⟶ ℤ
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
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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.
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
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