*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
*****************************************************************************
-----------------------------------------------------------------------------
Statistic identifier: St001885
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The number of binary words with the same proper border set.
The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$.
For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
@cached_function
def proper_border_set(w):
B = []
for i in range(1, len(w)):
if w[:i] == w[-i:]:
B.append(w[:i])
return B
def statistic(w):
B = proper_border_set(w)
return sum(1 for u in Words([0,1], len(w)) if B == proper_border_set(u))
-----------------------------------------------------------------------------
Statistic values:
0 => 2
1 => 2
00 => 1
01 => 2
10 => 2
11 => 1
000 => 1
001 => 4
010 => 1
011 => 4
100 => 4
101 => 1
110 => 4
111 => 1
0000 => 1
0001 => 6
0010 => 3
0011 => 6
0100 => 3
0101 => 1
0110 => 3
0111 => 6
1000 => 6
1001 => 3
1010 => 1
1011 => 3
1100 => 6
1101 => 3
1110 => 6
1111 => 1
00000 => 1
00001 => 12
00010 => 5
00011 => 12
00100 => 1
00101 => 12
00110 => 5
00111 => 12
01000 => 5
01001 => 2
01010 => 1
01011 => 12
01100 => 5
01101 => 2
01110 => 5
01111 => 12
10000 => 12
10001 => 5
10010 => 2
10011 => 5
10100 => 12
10101 => 1
10110 => 2
10111 => 5
11000 => 12
11001 => 5
11010 => 12
11011 => 1
11100 => 12
11101 => 5
11110 => 12
11111 => 1
000000 => 1
000001 => 20
000010 => 11
000011 => 20
000100 => 3
000101 => 20
000110 => 11
000111 => 20
001000 => 3
001001 => 1
001010 => 11
001011 => 20
001100 => 3
001101 => 20
001110 => 11
001111 => 20
010000 => 11
010001 => 3
010010 => 1
010011 => 20
010100 => 11
010101 => 1
010110 => 11
010111 => 20
011000 => 11
011001 => 3
011010 => 11
011011 => 1
011100 => 11
011101 => 3
011110 => 11
011111 => 20
100000 => 20
100001 => 11
100010 => 3
100011 => 11
100100 => 1
100101 => 11
100110 => 3
100111 => 11
101000 => 20
101001 => 11
101010 => 1
101011 => 11
101100 => 20
101101 => 1
101110 => 3
101111 => 11
110000 => 20
110001 => 11
110010 => 20
110011 => 3
110100 => 20
110101 => 11
110110 => 1
110111 => 3
111000 => 20
111001 => 11
111010 => 20
111011 => 3
111100 => 20
111101 => 11
111110 => 20
111111 => 1
0000000 => 1
0000001 => 40
0000010 => 19
0000011 => 40
0000100 => 6
0000101 => 40
0000110 => 19
0000111 => 40
0001000 => 1
0001001 => 40
0001010 => 19
0001011 => 40
0001100 => 6
0001101 => 40
0001110 => 19
0001111 => 40
0010000 => 6
0010001 => 2
0010010 => 1
0010011 => 40
0010100 => 6
0010101 => 40
0010110 => 19
0010111 => 40
0011000 => 6
0011001 => 2
0011010 => 19
0011011 => 40
0011100 => 6
0011101 => 40
0011110 => 19
0011111 => 40
0100000 => 19
0100001 => 8
0100010 => 1
0100011 => 40
0100100 => 1
0100101 => 8
0100110 => 19
0100111 => 40
0101000 => 19
0101001 => 8
0101010 => 1
0101011 => 40
0101100 => 19
0101101 => 8
0101110 => 19
0101111 => 40
0110000 => 19
0110001 => 8
0110010 => 19
0110011 => 2
0110100 => 19
0110101 => 8
0110110 => 1
0110111 => 40
0111000 => 19
0111001 => 8
0111010 => 19
0111011 => 2
0111100 => 19
0111101 => 8
0111110 => 19
0111111 => 40
1000000 => 40
1000001 => 19
1000010 => 8
1000011 => 19
1000100 => 2
1000101 => 19
1000110 => 8
1000111 => 19
1001000 => 40
1001001 => 1
1001010 => 8
1001011 => 19
1001100 => 2
1001101 => 19
1001110 => 8
1001111 => 19
1010000 => 40
1010001 => 19
1010010 => 8
1010011 => 19
1010100 => 40
1010101 => 1
1010110 => 8
1010111 => 19
1011000 => 40
1011001 => 19
1011010 => 8
1011011 => 1
1011100 => 40
1011101 => 1
1011110 => 8
1011111 => 19
1100000 => 40
1100001 => 19
1100010 => 40
1100011 => 6
1100100 => 40
1100101 => 19
1100110 => 2
1100111 => 6
1101000 => 40
1101001 => 19
1101010 => 40
1101011 => 6
1101100 => 40
1101101 => 1
1101110 => 2
1101111 => 6
1110000 => 40
1110001 => 19
1110010 => 40
1110011 => 6
1110100 => 40
1110101 => 19
1110110 => 40
1110111 => 1
1111000 => 40
1111001 => 19
1111010 => 40
1111011 => 6
1111100 => 40
1111101 => 19
1111110 => 40
1111111 => 1
00000000 => 1
00000001 => 74
00000010 => 41
00000011 => 74
00000100 => 11
00000101 => 74
00000110 => 41
00000111 => 74
00001000 => 3
00001001 => 74
00001010 => 41
00001011 => 74
00001100 => 11
00001101 => 74
00001110 => 41
00001111 => 74
00010000 => 3
00010001 => 1
00010010 => 41
00010011 => 74
00010100 => 11
00010101 => 74
00010110 => 41
00010111 => 74
00011000 => 3
00011001 => 74
00011010 => 41
00011011 => 74
00011100 => 11
00011101 => 74
00011110 => 41
00011111 => 74
00100000 => 11
00100001 => 4
00100010 => 1
00100011 => 74
00100100 => 1
00100101 => 74
00100110 => 41
00100111 => 74
00101000 => 11
00101001 => 4
00101010 => 41
00101011 => 74
00101100 => 11
00101101 => 74
00101110 => 41
00101111 => 74
00110000 => 11
00110001 => 4
00110010 => 41
00110011 => 1
00110100 => 11
00110101 => 74
00110110 => 41
00110111 => 74
00111000 => 11
00111001 => 4
00111010 => 41
00111011 => 74
00111100 => 11
00111101 => 74
00111110 => 41
00111111 => 74
01000000 => 41
01000001 => 13
01000010 => 4
01000011 => 74
01000100 => 1
01000101 => 13
01000110 => 41
01000111 => 74
01001000 => 41
01001001 => 1
01001010 => 4
01001011 => 74
01001100 => 41
01001101 => 13
01001110 => 41
01001111 => 74
01010000 => 41
01010001 => 13
01010010 => 4
01010011 => 74
01010100 => 41
01010101 => 1
01010110 => 41
01010111 => 74
01011000 => 41
01011001 => 13
01011010 => 4
01011011 => 74
01011100 => 41
01011101 => 13
01011110 => 41
01011111 => 74
01100000 => 41
01100001 => 13
01100010 => 41
01100011 => 4
01100100 => 41
01100101 => 13
01100110 => 1
01100111 => 74
01101000 => 41
01101001 => 13
01101010 => 41
01101011 => 4
01101100 => 41
01101101 => 1
01101110 => 41
01101111 => 74
01110000 => 41
01110001 => 13
01110010 => 41
01110011 => 4
01110100 => 41
01110101 => 13
01110110 => 41
01110111 => 1
01111000 => 41
01111001 => 13
01111010 => 41
01111011 => 4
01111100 => 41
01111101 => 13
01111110 => 41
01111111 => 74
10000000 => 74
10000001 => 41
10000010 => 13
10000011 => 41
10000100 => 4
10000101 => 41
10000110 => 13
10000111 => 41
10001000 => 1
10001001 => 41
10001010 => 13
10001011 => 41
10001100 => 4
10001101 => 41
10001110 => 13
10001111 => 41
10010000 => 74
10010001 => 41
10010010 => 1
10010011 => 41
10010100 => 4
10010101 => 41
10010110 => 13
10010111 => 41
10011000 => 74
10011001 => 1
10011010 => 13
10011011 => 41
10011100 => 4
10011101 => 41
10011110 => 13
10011111 => 41
10100000 => 74
10100001 => 41
10100010 => 13
10100011 => 41
10100100 => 74
10100101 => 4
10100110 => 13
10100111 => 41
10101000 => 74
10101001 => 41
10101010 => 1
10101011 => 41
10101100 => 74
10101101 => 4
10101110 => 13
10101111 => 41
10110000 => 74
10110001 => 41
10110010 => 13
10110011 => 41
10110100 => 74
10110101 => 4
10110110 => 1
10110111 => 41
10111000 => 74
10111001 => 41
10111010 => 13
10111011 => 1
10111100 => 74
10111101 => 4
10111110 => 13
10111111 => 41
11000000 => 74
11000001 => 41
11000010 => 74
11000011 => 11
11000100 => 74
11000101 => 41
11000110 => 4
11000111 => 11
11001000 => 74
11001001 => 41
11001010 => 74
11001011 => 11
11001100 => 1
11001101 => 41
11001110 => 4
11001111 => 11
11010000 => 74
11010001 => 41
11010010 => 74
11010011 => 11
11010100 => 74
11010101 => 41
11010110 => 4
11010111 => 11
11011000 => 74
11011001 => 41
11011010 => 74
11011011 => 1
11011100 => 74
11011101 => 1
11011110 => 4
11011111 => 11
11100000 => 74
11100001 => 41
11100010 => 74
11100011 => 11
11100100 => 74
11100101 => 41
11100110 => 74
11100111 => 3
11101000 => 74
11101001 => 41
11101010 => 74
11101011 => 11
11101100 => 74
11101101 => 41
11101110 => 1
11101111 => 3
11110000 => 74
11110001 => 41
11110010 => 74
11110011 => 11
11110100 => 74
11110101 => 41
11110110 => 74
11110111 => 3
11111000 => 74
11111001 => 41
11111010 => 74
11111011 => 11
11111100 => 74
11111101 => 41
11111110 => 74
11111111 => 1
000000000 => 1
000000001 => 148
000000010 => 77
000000011 => 148
000000100 => 23
000000101 => 148
000000110 => 77
000000111 => 148
000001000 => 6
000001001 => 148
000001010 => 77
000001011 => 148
000001100 => 23
000001101 => 148
000001110 => 77
000001111 => 148
000010000 => 1
000010001 => 148
000010010 => 77
000010011 => 148
000010100 => 23
000010101 => 148
000010110 => 77
000010111 => 148
000011000 => 6
000011001 => 148
000011010 => 77
000011011 => 148
000011100 => 23
000011101 => 148
000011110 => 77
000011111 => 148
000100000 => 6
000100001 => 2
000100010 => 1
000100011 => 148
000100100 => 23
000100101 => 148
000100110 => 77
000100111 => 148
000101000 => 6
000101001 => 148
000101010 => 77
000101011 => 148
000101100 => 23
000101101 => 148
000101110 => 77
000101111 => 148
000110000 => 6
000110001 => 2
000110010 => 77
000110011 => 148
000110100 => 23
000110101 => 148
000110110 => 77
000110111 => 148
000111000 => 6
000111001 => 148
000111010 => 77
000111011 => 148
000111100 => 23
000111101 => 148
000111110 => 77
000111111 => 148
001000000 => 23
001000001 => 7
001000010 => 2
001000011 => 148
001000100 => 1
001000101 => 148
001000110 => 77
001000111 => 148
001001000 => 23
001001001 => 1
001001010 => 77
001001011 => 148
001001100 => 23
001001101 => 148
001001110 => 77
001001111 => 148
001010000 => 23
001010001 => 7
001010010 => 2
001010011 => 148
001010100 => 23
001010101 => 148
001010110 => 77
001010111 => 148
001011000 => 23
001011001 => 7
001011010 => 77
001011011 => 148
001011100 => 23
001011101 => 148
001011110 => 77
001011111 => 148
001100000 => 23
001100001 => 7
001100010 => 77
001100011 => 2
001100100 => 23
001100101 => 148
001100110 => 1
001100111 => 148
001101000 => 23
001101001 => 7
001101010 => 77
001101011 => 148
001101100 => 23
001101101 => 148
001101110 => 77
001101111 => 148
001110000 => 23
001110001 => 7
001110010 => 77
001110011 => 2
001110100 => 23
001110101 => 148
001110110 => 77
001110111 => 148
001111000 => 23
001111001 => 7
001111010 => 77
001111011 => 148
001111100 => 23
001111101 => 148
001111110 => 77
001111111 => 148
010000000 => 77
010000001 => 30
010000010 => 6
010000011 => 148
010000100 => 2
010000101 => 30
010000110 => 77
010000111 => 148
010001000 => 1
010001001 => 30
010001010 => 6
010001011 => 148
010001100 => 77
010001101 => 30
010001110 => 77
010001111 => 148
010010000 => 77
010010001 => 30
010010010 => 1
010010011 => 148
010010100 => 2
010010101 => 30
010010110 => 77
010010111 => 148
010011000 => 77
010011001 => 30
010011010 => 6
010011011 => 148
010011100 => 77
010011101 => 30
010011110 => 77
010011111 => 148
010100000 => 77
010100001 => 30
010100010 => 6
010100011 => 148
010100100 => 77
010100101 => 2
010100110 => 77
010100111 => 148
010101000 => 77
010101001 => 30
010101010 => 1
010101011 => 148
010101100 => 77
010101101 => 30
010101110 => 77
010101111 => 148
010110000 => 77
010110001 => 30
010110010 => 6
010110011 => 148
010110100 => 77
010110101 => 2
010110110 => 77
010110111 => 148
010111000 => 77
010111001 => 30
010111010 => 6
010111011 => 148
010111100 => 77
010111101 => 30
010111110 => 77
010111111 => 148
011000000 => 77
011000001 => 30
011000010 => 77
011000011 => 7
011000100 => 77
011000101 => 30
011000110 => 2
011000111 => 148
011001000 => 77
011001001 => 30
011001010 => 77
011001011 => 7
011001100 => 1
011001101 => 30
011001110 => 77
011001111 => 148
011010000 => 77
011010001 => 30
011010010 => 77
011010011 => 7
011010100 => 77
011010101 => 30
011010110 => 2
011010111 => 148
011011000 => 77
011011001 => 30
011011010 => 77
011011011 => 1
011011100 => 77
011011101 => 30
011011110 => 77
011011111 => 148
011100000 => 77
011100001 => 30
011100010 => 77
011100011 => 7
011100100 => 77
011100101 => 30
011100110 => 77
011100111 => 2
011101000 => 77
011101001 => 30
011101010 => 77
011101011 => 7
011101100 => 77
011101101 => 30
011101110 => 1
011101111 => 148
011110000 => 77
011110001 => 30
011110010 => 77
011110011 => 7
011110100 => 77
011110101 => 30
011110110 => 77
011110111 => 2
011111000 => 77
011111001 => 30
011111010 => 77
011111011 => 7
011111100 => 77
011111101 => 30
011111110 => 77
011111111 => 148
100000000 => 148
100000001 => 77
100000010 => 30
100000011 => 77
100000100 => 7
100000101 => 77
100000110 => 30
100000111 => 77
100001000 => 2
100001001 => 77
100001010 => 30
100001011 => 77
100001100 => 7
100001101 => 77
100001110 => 30
100001111 => 77
100010000 => 148
100010001 => 1
100010010 => 30
100010011 => 77
100010100 => 7
100010101 => 77
100010110 => 30
100010111 => 77
100011000 => 2
100011001 => 77
100011010 => 30
100011011 => 77
100011100 => 7
100011101 => 77
100011110 => 30
100011111 => 77
100100000 => 148
100100001 => 77
100100010 => 30
100100011 => 77
100100100 => 1
100100101 => 77
100100110 => 30
100100111 => 77
100101000 => 148
100101001 => 2
100101010 => 30
100101011 => 77
100101100 => 7
100101101 => 77
100101110 => 30
100101111 => 77
100110000 => 148
100110001 => 77
100110010 => 30
100110011 => 1
100110100 => 7
100110101 => 77
100110110 => 30
100110111 => 77
100111000 => 148
100111001 => 2
100111010 => 30
100111011 => 77
100111100 => 7
100111101 => 77
100111110 => 30
100111111 => 77
101000000 => 148
101000001 => 77
101000010 => 30
101000011 => 77
101000100 => 148
101000101 => 6
101000110 => 30
101000111 => 77
101001000 => 148
101001001 => 77
101001010 => 2
101001011 => 77
101001100 => 148
101001101 => 6
101001110 => 30
101001111 => 77
101010000 => 148
101010001 => 77
101010010 => 30
101010011 => 77
101010100 => 148
101010101 => 1
101010110 => 30
101010111 => 77
101011000 => 148
101011001 => 77
101011010 => 2
101011011 => 77
101011100 => 148
101011101 => 6
101011110 => 30
101011111 => 77
101100000 => 148
101100001 => 77
101100010 => 30
101100011 => 77
101100100 => 148
101100101 => 6
101100110 => 30
101100111 => 77
101101000 => 148
101101001 => 77
101101010 => 30
101101011 => 2
101101100 => 148
101101101 => 1
101101110 => 30
101101111 => 77
101110000 => 148
101110001 => 77
101110010 => 30
101110011 => 77
101110100 => 148
101110101 => 6
101110110 => 30
101110111 => 1
101111000 => 148
101111001 => 77
101111010 => 30
101111011 => 2
101111100 => 148
101111101 => 6
101111110 => 30
101111111 => 77
110000000 => 148
110000001 => 77
110000010 => 148
110000011 => 23
110000100 => 148
110000101 => 77
110000110 => 7
110000111 => 23
110001000 => 148
110001001 => 77
110001010 => 148
110001011 => 23
110001100 => 2
110001101 => 77
110001110 => 7
110001111 => 23
110010000 => 148
110010001 => 77
110010010 => 148
110010011 => 23
110010100 => 148
110010101 => 77
110010110 => 7
110010111 => 23
110011000 => 148
110011001 => 1
110011010 => 148
110011011 => 23
110011100 => 2
110011101 => 77
110011110 => 7
110011111 => 23
110100000 => 148
110100001 => 77
110100010 => 148
110100011 => 23
110100100 => 148
110100101 => 77
110100110 => 7
110100111 => 23
110101000 => 148
110101001 => 77
110101010 => 148
110101011 => 23
110101100 => 148
110101101 => 2
110101110 => 7
110101111 => 23
110110000 => 148
110110001 => 77
110110010 => 148
110110011 => 23
110110100 => 148
110110101 => 77
110110110 => 1
110110111 => 23
110111000 => 148
110111001 => 77
110111010 => 148
110111011 => 1
110111100 => 148
110111101 => 2
110111110 => 7
110111111 => 23
111000000 => 148
111000001 => 77
111000010 => 148
111000011 => 23
111000100 => 148
111000101 => 77
111000110 => 148
111000111 => 6
111001000 => 148
111001001 => 77
111001010 => 148
111001011 => 23
111001100 => 148
111001101 => 77
111001110 => 2
111001111 => 6
111010000 => 148
111010001 => 77
111010010 => 148
111010011 => 23
111010100 => 148
111010101 => 77
111010110 => 148
111010111 => 6
111011000 => 148
111011001 => 77
111011010 => 148
111011011 => 23
111011100 => 148
111011101 => 1
111011110 => 2
111011111 => 6
111100000 => 148
111100001 => 77
111100010 => 148
111100011 => 23
111100100 => 148
111100101 => 77
111100110 => 148
111100111 => 6
111101000 => 148
111101001 => 77
111101010 => 148
111101011 => 23
111101100 => 148
111101101 => 77
111101110 => 148
111101111 => 1
111110000 => 148
111110001 => 77
111110010 => 148
111110011 => 23
111110100 => 148
111110101 => 77
111110110 => 148
111110111 => 6
111111000 => 148
111111001 => 77
111111010 => 148
111111011 => 23
111111100 => 148
111111101 => 77
111111110 => 148
111111111 => 1
-----------------------------------------------------------------------------
Created: Dec 21, 2022 at 11:43 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Dec 21, 2022 at 11:43 by Martin Rubey