*****************************************************************************
* 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: St001355
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: Number of non-empty prefixes of a binary word that contain equally many 0's and 1's.
Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(P):
return sum(1 for i in range(2,len(P)+1,2) if P[:i].count(0) == P[:i].count(1) )
-----------------------------------------------------------------------------
Statistic values:
0 => 0
1 => 0
00 => 0
01 => 1
10 => 1
11 => 0
000 => 0
001 => 0
010 => 1
011 => 1
100 => 1
101 => 1
110 => 0
111 => 0
0000 => 0
0001 => 0
0010 => 0
0011 => 1
0100 => 1
0101 => 2
0110 => 2
0111 => 1
1000 => 1
1001 => 2
1010 => 2
1011 => 1
1100 => 1
1101 => 0
1110 => 0
1111 => 0
00000 => 0
00001 => 0
00010 => 0
00011 => 0
00100 => 0
00101 => 0
00110 => 1
00111 => 1
01000 => 1
01001 => 1
01010 => 2
01011 => 2
01100 => 2
01101 => 2
01110 => 1
01111 => 1
10000 => 1
10001 => 1
10010 => 2
10011 => 2
10100 => 2
10101 => 2
10110 => 1
10111 => 1
11000 => 1
11001 => 1
11010 => 0
11011 => 0
11100 => 0
11101 => 0
11110 => 0
11111 => 0
000000 => 0
000001 => 0
000010 => 0
000011 => 0
000100 => 0
000101 => 0
000110 => 0
000111 => 1
001000 => 0
001001 => 0
001010 => 0
001011 => 1
001100 => 1
001101 => 2
001110 => 2
001111 => 1
010000 => 1
010001 => 1
010010 => 1
010011 => 2
010100 => 2
010101 => 3
010110 => 3
010111 => 2
011000 => 2
011001 => 3
011010 => 3
011011 => 2
011100 => 2
011101 => 1
011110 => 1
011111 => 1
100000 => 1
100001 => 1
100010 => 1
100011 => 2
100100 => 2
100101 => 3
100110 => 3
100111 => 2
101000 => 2
101001 => 3
101010 => 3
101011 => 2
101100 => 2
101101 => 1
101110 => 1
101111 => 1
110000 => 1
110001 => 2
110010 => 2
110011 => 1
110100 => 1
110101 => 0
110110 => 0
110111 => 0
111000 => 1
111001 => 0
111010 => 0
111011 => 0
111100 => 0
111101 => 0
111110 => 0
111111 => 0
0000000 => 0
0000001 => 0
0000010 => 0
0000011 => 0
0000100 => 0
0000101 => 0
0000110 => 0
0000111 => 0
0001000 => 0
0001001 => 0
0001010 => 0
0001011 => 0
0001100 => 0
0001101 => 0
0001110 => 1
0001111 => 1
0010000 => 0
0010001 => 0
0010010 => 0
0010011 => 0
0010100 => 0
0010101 => 0
0010110 => 1
0010111 => 1
0011000 => 1
0011001 => 1
0011010 => 2
0011011 => 2
0011100 => 2
0011101 => 2
0011110 => 1
0011111 => 1
0100000 => 1
0100001 => 1
0100010 => 1
0100011 => 1
0100100 => 1
0100101 => 1
0100110 => 2
0100111 => 2
0101000 => 2
0101001 => 2
0101010 => 3
0101011 => 3
0101100 => 3
0101101 => 3
0101110 => 2
0101111 => 2
0110000 => 2
0110001 => 2
0110010 => 3
0110011 => 3
0110100 => 3
0110101 => 3
0110110 => 2
0110111 => 2
0111000 => 2
0111001 => 2
0111010 => 1
0111011 => 1
0111100 => 1
0111101 => 1
0111110 => 1
0111111 => 1
1000000 => 1
1000001 => 1
1000010 => 1
1000011 => 1
1000100 => 1
1000101 => 1
1000110 => 2
1000111 => 2
1001000 => 2
1001001 => 2
1001010 => 3
1001011 => 3
1001100 => 3
1001101 => 3
1001110 => 2
1001111 => 2
1010000 => 2
1010001 => 2
1010010 => 3
1010011 => 3
1010100 => 3
1010101 => 3
1010110 => 2
1010111 => 2
1011000 => 2
1011001 => 2
1011010 => 1
1011011 => 1
1011100 => 1
1011101 => 1
1011110 => 1
1011111 => 1
1100000 => 1
1100001 => 1
1100010 => 2
1100011 => 2
1100100 => 2
1100101 => 2
1100110 => 1
1100111 => 1
1101000 => 1
1101001 => 1
1101010 => 0
1101011 => 0
1101100 => 0
1101101 => 0
1101110 => 0
1101111 => 0
1110000 => 1
1110001 => 1
1110010 => 0
1110011 => 0
1110100 => 0
1110101 => 0
1110110 => 0
1110111 => 0
1111000 => 0
1111001 => 0
1111010 => 0
1111011 => 0
1111100 => 0
1111101 => 0
1111110 => 0
1111111 => 0
00000000 => 0
00000001 => 0
00000010 => 0
00000011 => 0
00000100 => 0
00000101 => 0
00000110 => 0
00000111 => 0
00001000 => 0
00001001 => 0
00001010 => 0
00001011 => 0
00001100 => 0
00001101 => 0
00001110 => 0
00001111 => 1
00010000 => 0
00010001 => 0
00010010 => 0
00010011 => 0
00010100 => 0
00010101 => 0
00010110 => 0
00010111 => 1
00011000 => 0
00011001 => 0
00011010 => 0
00011011 => 1
00011100 => 1
00011101 => 2
00011110 => 2
00011111 => 1
00100000 => 0
00100001 => 0
00100010 => 0
00100011 => 0
00100100 => 0
00100101 => 0
00100110 => 0
00100111 => 1
00101000 => 0
00101001 => 0
00101010 => 0
00101011 => 1
00101100 => 1
00101101 => 2
00101110 => 2
00101111 => 1
00110000 => 1
00110001 => 1
00110010 => 1
00110011 => 2
00110100 => 2
00110101 => 3
00110110 => 3
00110111 => 2
00111000 => 2
00111001 => 3
00111010 => 3
00111011 => 2
00111100 => 2
00111101 => 1
00111110 => 1
00111111 => 1
01000000 => 1
01000001 => 1
01000010 => 1
01000011 => 1
01000100 => 1
01000101 => 1
01000110 => 1
01000111 => 2
01001000 => 1
01001001 => 1
01001010 => 1
01001011 => 2
01001100 => 2
01001101 => 3
01001110 => 3
01001111 => 2
01010000 => 2
01010001 => 2
01010010 => 2
01010011 => 3
01010100 => 3
01010101 => 4
01010110 => 4
01010111 => 3
01011000 => 3
01011001 => 4
01011010 => 4
01011011 => 3
01011100 => 3
01011101 => 2
01011110 => 2
01011111 => 2
01100000 => 2
01100001 => 2
01100010 => 2
01100011 => 3
01100100 => 3
01100101 => 4
01100110 => 4
01100111 => 3
01101000 => 3
01101001 => 4
01101010 => 4
01101011 => 3
01101100 => 3
01101101 => 2
01101110 => 2
01101111 => 2
01110000 => 2
01110001 => 3
01110010 => 3
01110011 => 2
01110100 => 2
01110101 => 1
01110110 => 1
01110111 => 1
01111000 => 2
01111001 => 1
01111010 => 1
01111011 => 1
01111100 => 1
01111101 => 1
01111110 => 1
01111111 => 1
10000000 => 1
10000001 => 1
10000010 => 1
10000011 => 1
10000100 => 1
10000101 => 1
10000110 => 1
10000111 => 2
10001000 => 1
10001001 => 1
10001010 => 1
10001011 => 2
10001100 => 2
10001101 => 3
10001110 => 3
10001111 => 2
10010000 => 2
10010001 => 2
10010010 => 2
10010011 => 3
10010100 => 3
10010101 => 4
10010110 => 4
10010111 => 3
10011000 => 3
10011001 => 4
10011010 => 4
10011011 => 3
10011100 => 3
10011101 => 2
10011110 => 2
10011111 => 2
10100000 => 2
10100001 => 2
10100010 => 2
10100011 => 3
10100100 => 3
10100101 => 4
10100110 => 4
10100111 => 3
10101000 => 3
10101001 => 4
10101010 => 4
10101011 => 3
10101100 => 3
10101101 => 2
10101110 => 2
10101111 => 2
10110000 => 2
10110001 => 3
10110010 => 3
10110011 => 2
10110100 => 2
10110101 => 1
10110110 => 1
10110111 => 1
10111000 => 2
10111001 => 1
10111010 => 1
10111011 => 1
10111100 => 1
10111101 => 1
10111110 => 1
10111111 => 1
11000000 => 1
11000001 => 1
11000010 => 1
11000011 => 2
11000100 => 2
11000101 => 3
11000110 => 3
11000111 => 2
11001000 => 2
11001001 => 3
11001010 => 3
11001011 => 2
11001100 => 2
11001101 => 1
11001110 => 1
11001111 => 1
11010000 => 1
11010001 => 2
11010010 => 2
11010011 => 1
11010100 => 1
11010101 => 0
11010110 => 0
11010111 => 0
11011000 => 1
11011001 => 0
11011010 => 0
11011011 => 0
11011100 => 0
11011101 => 0
11011110 => 0
11011111 => 0
11100000 => 1
11100001 => 2
11100010 => 2
11100011 => 1
11100100 => 1
11100101 => 0
11100110 => 0
11100111 => 0
11101000 => 1
11101001 => 0
11101010 => 0
11101011 => 0
11101100 => 0
11101101 => 0
11101110 => 0
11101111 => 0
11110000 => 1
11110001 => 0
11110010 => 0
11110011 => 0
11110100 => 0
11110101 => 0
11110110 => 0
11110111 => 0
11111000 => 0
11111001 => 0
11111010 => 0
11111011 => 0
11111100 => 0
11111101 => 0
11111110 => 0
11111111 => 0
000000000 => 0
000000001 => 0
000000010 => 0
000000011 => 0
000000100 => 0
000000101 => 0
000000110 => 0
000000111 => 0
000001000 => 0
000001001 => 0
000001010 => 0
000001011 => 0
000001100 => 0
000001101 => 0
000001110 => 0
000001111 => 0
000010000 => 0
000010001 => 0
000010010 => 0
000010011 => 0
000010100 => 0
000010101 => 0
000010110 => 0
000010111 => 0
000011000 => 0
000011001 => 0
000011010 => 0
000011011 => 0
000011100 => 0
000011101 => 0
000011110 => 1
000011111 => 1
000100000 => 0
000100001 => 0
000100010 => 0
000100011 => 0
000100100 => 0
000100101 => 0
000100110 => 0
000100111 => 0
000101000 => 0
000101001 => 0
000101010 => 0
000101011 => 0
000101100 => 0
000101101 => 0
000101110 => 1
000101111 => 1
000110000 => 0
000110001 => 0
000110010 => 0
000110011 => 0
000110100 => 0
000110101 => 0
000110110 => 1
000110111 => 1
000111000 => 1
000111001 => 1
000111010 => 2
000111011 => 2
000111100 => 2
000111101 => 2
000111110 => 1
000111111 => 1
001000000 => 0
001000001 => 0
001000010 => 0
001000011 => 0
001000100 => 0
001000101 => 0
001000110 => 0
001000111 => 0
001001000 => 0
001001001 => 0
001001010 => 0
001001011 => 0
001001100 => 0
001001101 => 0
001001110 => 1
001001111 => 1
001010000 => 0
001010001 => 0
001010010 => 0
001010011 => 0
001010100 => 0
001010101 => 0
001010110 => 1
001010111 => 1
001011000 => 1
001011001 => 1
001011010 => 2
001011011 => 2
001011100 => 2
001011101 => 2
001011110 => 1
001011111 => 1
001100000 => 1
001100001 => 1
001100010 => 1
001100011 => 1
001100100 => 1
001100101 => 1
001100110 => 2
001100111 => 2
001101000 => 2
001101001 => 2
001101010 => 3
001101011 => 3
001101100 => 3
001101101 => 3
001101110 => 2
001101111 => 2
001110000 => 2
001110001 => 2
001110010 => 3
001110011 => 3
001110100 => 3
001110101 => 3
001110110 => 2
001110111 => 2
001111000 => 2
001111001 => 2
001111010 => 1
001111011 => 1
001111100 => 1
001111101 => 1
001111110 => 1
001111111 => 1
010000000 => 1
010000001 => 1
010000010 => 1
010000011 => 1
010000100 => 1
010000101 => 1
010000110 => 1
010000111 => 1
010001000 => 1
010001001 => 1
010001010 => 1
010001011 => 1
010001100 => 1
010001101 => 1
010001110 => 2
010001111 => 2
010010000 => 1
010010001 => 1
010010010 => 1
010010011 => 1
010010100 => 1
010010101 => 1
010010110 => 2
010010111 => 2
010011000 => 2
010011001 => 2
010011010 => 3
010011011 => 3
010011100 => 3
010011101 => 3
010011110 => 2
010011111 => 2
010100000 => 2
010100001 => 2
010100010 => 2
010100011 => 2
010100100 => 2
010100101 => 2
010100110 => 3
010100111 => 3
010101000 => 3
010101001 => 3
010101010 => 4
010101011 => 4
010101100 => 4
010101101 => 4
010101110 => 3
010101111 => 3
010110000 => 3
010110001 => 3
010110010 => 4
010110011 => 4
010110100 => 4
010110101 => 4
010110110 => 3
010110111 => 3
010111000 => 3
010111001 => 3
010111010 => 2
010111011 => 2
010111100 => 2
010111101 => 2
010111110 => 2
010111111 => 2
011000000 => 2
011000001 => 2
011000010 => 2
011000011 => 2
011000100 => 2
011000101 => 2
011000110 => 3
011000111 => 3
011001000 => 3
011001001 => 3
011001010 => 4
011001011 => 4
011001100 => 4
011001101 => 4
011001110 => 3
011001111 => 3
011010000 => 3
011010001 => 3
011010010 => 4
011010011 => 4
011010100 => 4
011010101 => 4
011010110 => 3
011010111 => 3
011011000 => 3
011011001 => 3
011011010 => 2
011011011 => 2
011011100 => 2
011011101 => 2
011011110 => 2
011011111 => 2
011100000 => 2
011100001 => 2
011100010 => 3
011100011 => 3
011100100 => 3
011100101 => 3
011100110 => 2
011100111 => 2
011101000 => 2
011101001 => 2
011101010 => 1
011101011 => 1
011101100 => 1
011101101 => 1
011101110 => 1
011101111 => 1
011110000 => 2
011110001 => 2
011110010 => 1
011110011 => 1
011110100 => 1
011110101 => 1
011110110 => 1
011110111 => 1
011111000 => 1
011111001 => 1
011111010 => 1
011111011 => 1
011111100 => 1
011111101 => 1
011111110 => 1
011111111 => 1
100000000 => 1
100000001 => 1
100000010 => 1
100000011 => 1
100000100 => 1
100000101 => 1
100000110 => 1
100000111 => 1
100001000 => 1
100001001 => 1
100001010 => 1
100001011 => 1
100001100 => 1
100001101 => 1
100001110 => 2
100001111 => 2
100010000 => 1
100010001 => 1
100010010 => 1
100010011 => 1
100010100 => 1
100010101 => 1
100010110 => 2
100010111 => 2
100011000 => 2
100011001 => 2
100011010 => 3
100011011 => 3
100011100 => 3
100011101 => 3
100011110 => 2
100011111 => 2
100100000 => 2
100100001 => 2
100100010 => 2
100100011 => 2
100100100 => 2
100100101 => 2
100100110 => 3
100100111 => 3
100101000 => 3
100101001 => 3
100101010 => 4
100101011 => 4
100101100 => 4
100101101 => 4
100101110 => 3
100101111 => 3
100110000 => 3
100110001 => 3
100110010 => 4
100110011 => 4
100110100 => 4
100110101 => 4
100110110 => 3
100110111 => 3
100111000 => 3
100111001 => 3
100111010 => 2
100111011 => 2
100111100 => 2
100111101 => 2
100111110 => 2
100111111 => 2
101000000 => 2
101000001 => 2
101000010 => 2
101000011 => 2
101000100 => 2
101000101 => 2
101000110 => 3
101000111 => 3
101001000 => 3
101001001 => 3
101001010 => 4
101001011 => 4
101001100 => 4
101001101 => 4
101001110 => 3
101001111 => 3
101010000 => 3
101010001 => 3
101010010 => 4
101010011 => 4
101010100 => 4
101010101 => 4
101010110 => 3
101010111 => 3
101011000 => 3
101011001 => 3
101011010 => 2
101011011 => 2
101011100 => 2
101011101 => 2
101011110 => 2
101011111 => 2
101100000 => 2
101100001 => 2
101100010 => 3
101100011 => 3
101100100 => 3
101100101 => 3
101100110 => 2
101100111 => 2
101101000 => 2
101101001 => 2
101101010 => 1
101101011 => 1
101101100 => 1
101101101 => 1
101101110 => 1
101101111 => 1
101110000 => 2
101110001 => 2
101110010 => 1
101110011 => 1
101110100 => 1
101110101 => 1
101110110 => 1
101110111 => 1
101111000 => 1
101111001 => 1
101111010 => 1
101111011 => 1
101111100 => 1
101111101 => 1
101111110 => 1
101111111 => 1
110000000 => 1
110000001 => 1
110000010 => 1
110000011 => 1
110000100 => 1
110000101 => 1
110000110 => 2
110000111 => 2
110001000 => 2
110001001 => 2
110001010 => 3
110001011 => 3
110001100 => 3
110001101 => 3
110001110 => 2
110001111 => 2
110010000 => 2
110010001 => 2
110010010 => 3
110010011 => 3
110010100 => 3
110010101 => 3
110010110 => 2
110010111 => 2
110011000 => 2
110011001 => 2
110011010 => 1
110011011 => 1
110011100 => 1
110011101 => 1
110011110 => 1
110011111 => 1
110100000 => 1
110100001 => 1
110100010 => 2
110100011 => 2
110100100 => 2
110100101 => 2
110100110 => 1
110100111 => 1
110101000 => 1
110101001 => 1
110101010 => 0
110101011 => 0
110101100 => 0
110101101 => 0
110101110 => 0
110101111 => 0
110110000 => 1
110110001 => 1
110110010 => 0
110110011 => 0
110110100 => 0
110110101 => 0
110110110 => 0
110110111 => 0
110111000 => 0
110111001 => 0
110111010 => 0
110111011 => 0
110111100 => 0
110111101 => 0
110111110 => 0
110111111 => 0
111000000 => 1
111000001 => 1
111000010 => 2
111000011 => 2
111000100 => 2
111000101 => 2
111000110 => 1
111000111 => 1
111001000 => 1
111001001 => 1
111001010 => 0
111001011 => 0
111001100 => 0
111001101 => 0
111001110 => 0
111001111 => 0
111010000 => 1
111010001 => 1
111010010 => 0
111010011 => 0
111010100 => 0
111010101 => 0
111010110 => 0
111010111 => 0
111011000 => 0
111011001 => 0
111011010 => 0
111011011 => 0
111011100 => 0
111011101 => 0
111011110 => 0
111011111 => 0
111100000 => 1
111100001 => 1
111100010 => 0
111100011 => 0
111100100 => 0
111100101 => 0
111100110 => 0
111100111 => 0
111101000 => 0
111101001 => 0
111101010 => 0
111101011 => 0
111101100 => 0
111101101 => 0
111101110 => 0
111101111 => 0
111110000 => 0
111110001 => 0
111110010 => 0
111110011 => 0
111110100 => 0
111110101 => 0
111110110 => 0
111110111 => 0
111111000 => 0
111111001 => 0
111111010 => 0
111111011 => 0
111111100 => 0
111111101 => 0
111111110 => 0
111111111 => 0
-----------------------------------------------------------------------------
Created: Feb 11, 2019 at 18:21 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Feb 12, 2019 at 07:43 by Christian Stump