*****************************************************************************
* 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: St001730
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
-----------------------------------------------------------------------------
References: [1] Elizalde, S. Counting lattice paths by crossings and major index I: the corner-flipping bijections [[arXiv:2106.09878]]
-----------------------------------------------------------------------------
Code:
def statistic(w):
c = 0
h = 0
for i in range(len(w)-1):
if ((h == 1 and w[i] == w[i+1] == 0) or
(h == -1 and w[i] == w[i+1] == 1)):
c += 1
if w[i] == 0:
h -= 1
else:
h += 1
return c
-----------------------------------------------------------------------------
Statistic values:
0 => 0
1 => 0
00 => 0
01 => 0
10 => 0
11 => 0
000 => 0
001 => 0
010 => 0
011 => 1
100 => 1
101 => 0
110 => 0
111 => 0
0000 => 0
0001 => 0
0010 => 0
0011 => 0
0100 => 0
0101 => 0
0110 => 1
0111 => 1
1000 => 1
1001 => 1
1010 => 0
1011 => 0
1100 => 0
1101 => 0
1110 => 0
1111 => 0
00000 => 0
00001 => 0
00010 => 0
00011 => 0
00100 => 0
00101 => 0
00110 => 0
00111 => 1
01000 => 0
01001 => 0
01010 => 0
01011 => 1
01100 => 2
01101 => 1
01110 => 1
01111 => 1
10000 => 1
10001 => 1
10010 => 1
10011 => 2
10100 => 1
10101 => 0
10110 => 0
10111 => 0
11000 => 1
11001 => 0
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 => 0
001000 => 0
001001 => 0
001010 => 0
001011 => 0
001100 => 0
001101 => 0
001110 => 1
001111 => 1
010000 => 0
010001 => 0
010010 => 0
010011 => 0
010100 => 0
010101 => 0
010110 => 1
010111 => 1
011000 => 2
011001 => 2
011010 => 1
011011 => 1
011100 => 1
011101 => 1
011110 => 1
011111 => 1
100000 => 1
100001 => 1
100010 => 1
100011 => 1
100100 => 1
100101 => 1
100110 => 2
100111 => 2
101000 => 1
101001 => 1
101010 => 0
101011 => 0
101100 => 0
101101 => 0
101110 => 0
101111 => 0
110000 => 1
110001 => 1
110010 => 0
110011 => 0
110100 => 0
110101 => 0
110110 => 0
110111 => 0
111000 => 0
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 => 0
0001111 => 1
0010000 => 0
0010001 => 0
0010010 => 0
0010011 => 0
0010100 => 0
0010101 => 0
0010110 => 0
0010111 => 1
0011000 => 0
0011001 => 0
0011010 => 0
0011011 => 1
0011100 => 2
0011101 => 1
0011110 => 1
0011111 => 1
0100000 => 0
0100001 => 0
0100010 => 0
0100011 => 0
0100100 => 0
0100101 => 0
0100110 => 0
0100111 => 1
0101000 => 0
0101001 => 0
0101010 => 0
0101011 => 1
0101100 => 2
0101101 => 1
0101110 => 1
0101111 => 1
0110000 => 2
0110001 => 2
0110010 => 2
0110011 => 3
0110100 => 2
0110101 => 1
0110110 => 1
0110111 => 1
0111000 => 2
0111001 => 1
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 => 1
1000111 => 2
1001000 => 1
1001001 => 1
1001010 => 1
1001011 => 2
1001100 => 3
1001101 => 2
1001110 => 2
1001111 => 2
1010000 => 1
1010001 => 1
1010010 => 1
1010011 => 2
1010100 => 1
1010101 => 0
1010110 => 0
1010111 => 0
1011000 => 1
1011001 => 0
1011010 => 0
1011011 => 0
1011100 => 0
1011101 => 0
1011110 => 0
1011111 => 0
1100000 => 1
1100001 => 1
1100010 => 1
1100011 => 2
1100100 => 1
1100101 => 0
1100110 => 0
1100111 => 0
1101000 => 1
1101001 => 0
1101010 => 0
1101011 => 0
1101100 => 0
1101101 => 0
1101110 => 0
1101111 => 0
1110000 => 1
1110001 => 0
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 => 0
00010000 => 0
00010001 => 0
00010010 => 0
00010011 => 0
00010100 => 0
00010101 => 0
00010110 => 0
00010111 => 0
00011000 => 0
00011001 => 0
00011010 => 0
00011011 => 0
00011100 => 0
00011101 => 0
00011110 => 1
00011111 => 1
00100000 => 0
00100001 => 0
00100010 => 0
00100011 => 0
00100100 => 0
00100101 => 0
00100110 => 0
00100111 => 0
00101000 => 0
00101001 => 0
00101010 => 0
00101011 => 0
00101100 => 0
00101101 => 0
00101110 => 1
00101111 => 1
00110000 => 0
00110001 => 0
00110010 => 0
00110011 => 0
00110100 => 0
00110101 => 0
00110110 => 1
00110111 => 1
00111000 => 2
00111001 => 2
00111010 => 1
00111011 => 1
00111100 => 1
00111101 => 1
00111110 => 1
00111111 => 1
01000000 => 0
01000001 => 0
01000010 => 0
01000011 => 0
01000100 => 0
01000101 => 0
01000110 => 0
01000111 => 0
01001000 => 0
01001001 => 0
01001010 => 0
01001011 => 0
01001100 => 0
01001101 => 0
01001110 => 1
01001111 => 1
01010000 => 0
01010001 => 0
01010010 => 0
01010011 => 0
01010100 => 0
01010101 => 0
01010110 => 1
01010111 => 1
01011000 => 2
01011001 => 2
01011010 => 1
01011011 => 1
01011100 => 1
01011101 => 1
01011110 => 1
01011111 => 1
01100000 => 2
01100001 => 2
01100010 => 2
01100011 => 2
01100100 => 2
01100101 => 2
01100110 => 3
01100111 => 3
01101000 => 2
01101001 => 2
01101010 => 1
01101011 => 1
01101100 => 1
01101101 => 1
01101110 => 1
01101111 => 1
01110000 => 2
01110001 => 2
01110010 => 1
01110011 => 1
01110100 => 1
01110101 => 1
01110110 => 1
01110111 => 1
01111000 => 1
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 => 1
10001000 => 1
10001001 => 1
10001010 => 1
10001011 => 1
10001100 => 1
10001101 => 1
10001110 => 2
10001111 => 2
10010000 => 1
10010001 => 1
10010010 => 1
10010011 => 1
10010100 => 1
10010101 => 1
10010110 => 2
10010111 => 2
10011000 => 3
10011001 => 3
10011010 => 2
10011011 => 2
10011100 => 2
10011101 => 2
10011110 => 2
10011111 => 2
10100000 => 1
10100001 => 1
10100010 => 1
10100011 => 1
10100100 => 1
10100101 => 1
10100110 => 2
10100111 => 2
10101000 => 1
10101001 => 1
10101010 => 0
10101011 => 0
10101100 => 0
10101101 => 0
10101110 => 0
10101111 => 0
10110000 => 1
10110001 => 1
10110010 => 0
10110011 => 0
10110100 => 0
10110101 => 0
10110110 => 0
10110111 => 0
10111000 => 0
10111001 => 0
10111010 => 0
10111011 => 0
10111100 => 0
10111101 => 0
10111110 => 0
10111111 => 0
11000000 => 1
11000001 => 1
11000010 => 1
11000011 => 1
11000100 => 1
11000101 => 1
11000110 => 2
11000111 => 2
11001000 => 1
11001001 => 1
11001010 => 0
11001011 => 0
11001100 => 0
11001101 => 0
11001110 => 0
11001111 => 0
11010000 => 1
11010001 => 1
11010010 => 0
11010011 => 0
11010100 => 0
11010101 => 0
11010110 => 0
11010111 => 0
11011000 => 0
11011001 => 0
11011010 => 0
11011011 => 0
11011100 => 0
11011101 => 0
11011110 => 0
11011111 => 0
11100000 => 1
11100001 => 1
11100010 => 0
11100011 => 0
11100100 => 0
11100101 => 0
11100110 => 0
11100111 => 0
11101000 => 0
11101001 => 0
11101010 => 0
11101011 => 0
11101100 => 0
11101101 => 0
11101110 => 0
11101111 => 0
11110000 => 0
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 => 0
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 => 0
000101111 => 1
000110000 => 0
000110001 => 0
000110010 => 0
000110011 => 0
000110100 => 0
000110101 => 0
000110110 => 0
000110111 => 1
000111000 => 0
000111001 => 0
000111010 => 0
000111011 => 1
000111100 => 2
000111101 => 1
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 => 0
001001111 => 1
001010000 => 0
001010001 => 0
001010010 => 0
001010011 => 0
001010100 => 0
001010101 => 0
001010110 => 0
001010111 => 1
001011000 => 0
001011001 => 0
001011010 => 0
001011011 => 1
001011100 => 2
001011101 => 1
001011110 => 1
001011111 => 1
001100000 => 0
001100001 => 0
001100010 => 0
001100011 => 0
001100100 => 0
001100101 => 0
001100110 => 0
001100111 => 1
001101000 => 0
001101001 => 0
001101010 => 0
001101011 => 1
001101100 => 2
001101101 => 1
001101110 => 1
001101111 => 1
001110000 => 2
001110001 => 2
001110010 => 2
001110011 => 3
001110100 => 2
001110101 => 1
001110110 => 1
001110111 => 1
001111000 => 2
001111001 => 1
001111010 => 1
001111011 => 1
001111100 => 1
001111101 => 1
001111110 => 1
001111111 => 1
010000000 => 0
010000001 => 0
010000010 => 0
010000011 => 0
010000100 => 0
010000101 => 0
010000110 => 0
010000111 => 0
010001000 => 0
010001001 => 0
010001010 => 0
010001011 => 0
010001100 => 0
010001101 => 0
010001110 => 0
010001111 => 1
010010000 => 0
010010001 => 0
010010010 => 0
010010011 => 0
010010100 => 0
010010101 => 0
010010110 => 0
010010111 => 1
010011000 => 0
010011001 => 0
010011010 => 0
010011011 => 1
010011100 => 2
010011101 => 1
010011110 => 1
010011111 => 1
010100000 => 0
010100001 => 0
010100010 => 0
010100011 => 0
010100100 => 0
010100101 => 0
010100110 => 0
010100111 => 1
010101000 => 0
010101001 => 0
010101010 => 0
010101011 => 1
010101100 => 2
010101101 => 1
010101110 => 1
010101111 => 1
010110000 => 2
010110001 => 2
010110010 => 2
010110011 => 3
010110100 => 2
010110101 => 1
010110110 => 1
010110111 => 1
010111000 => 2
010111001 => 1
010111010 => 1
010111011 => 1
010111100 => 1
010111101 => 1
010111110 => 1
010111111 => 1
011000000 => 2
011000001 => 2
011000010 => 2
011000011 => 2
011000100 => 2
011000101 => 2
011000110 => 2
011000111 => 3
011001000 => 2
011001001 => 2
011001010 => 2
011001011 => 3
011001100 => 4
011001101 => 3
011001110 => 3
011001111 => 3
011010000 => 2
011010001 => 2
011010010 => 2
011010011 => 3
011010100 => 2
011010101 => 1
011010110 => 1
011010111 => 1
011011000 => 2
011011001 => 1
011011010 => 1
011011011 => 1
011011100 => 1
011011101 => 1
011011110 => 1
011011111 => 1
011100000 => 2
011100001 => 2
011100010 => 2
011100011 => 3
011100100 => 2
011100101 => 1
011100110 => 1
011100111 => 1
011101000 => 2
011101001 => 1
011101010 => 1
011101011 => 1
011101100 => 1
011101101 => 1
011101110 => 1
011101111 => 1
011110000 => 2
011110001 => 1
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 => 1
100001111 => 2
100010000 => 1
100010001 => 1
100010010 => 1
100010011 => 1
100010100 => 1
100010101 => 1
100010110 => 1
100010111 => 2
100011000 => 1
100011001 => 1
100011010 => 1
100011011 => 2
100011100 => 3
100011101 => 2
100011110 => 2
100011111 => 2
100100000 => 1
100100001 => 1
100100010 => 1
100100011 => 1
100100100 => 1
100100101 => 1
100100110 => 1
100100111 => 2
100101000 => 1
100101001 => 1
100101010 => 1
100101011 => 2
100101100 => 3
100101101 => 2
100101110 => 2
100101111 => 2
100110000 => 3
100110001 => 3
100110010 => 3
100110011 => 4
100110100 => 3
100110101 => 2
100110110 => 2
100110111 => 2
100111000 => 3
100111001 => 2
100111010 => 2
100111011 => 2
100111100 => 2
100111101 => 2
100111110 => 2
100111111 => 2
101000000 => 1
101000001 => 1
101000010 => 1
101000011 => 1
101000100 => 1
101000101 => 1
101000110 => 1
101000111 => 2
101001000 => 1
101001001 => 1
101001010 => 1
101001011 => 2
101001100 => 3
101001101 => 2
101001110 => 2
101001111 => 2
101010000 => 1
101010001 => 1
101010010 => 1
101010011 => 2
101010100 => 1
101010101 => 0
101010110 => 0
101010111 => 0
101011000 => 1
101011001 => 0
101011010 => 0
101011011 => 0
101011100 => 0
101011101 => 0
101011110 => 0
101011111 => 0
101100000 => 1
101100001 => 1
101100010 => 1
101100011 => 2
101100100 => 1
101100101 => 0
101100110 => 0
101100111 => 0
101101000 => 1
101101001 => 0
101101010 => 0
101101011 => 0
101101100 => 0
101101101 => 0
101101110 => 0
101101111 => 0
101110000 => 1
101110001 => 0
101110010 => 0
101110011 => 0
101110100 => 0
101110101 => 0
101110110 => 0
101110111 => 0
101111000 => 0
101111001 => 0
101111010 => 0
101111011 => 0
101111100 => 0
101111101 => 0
101111110 => 0
101111111 => 0
110000000 => 1
110000001 => 1
110000010 => 1
110000011 => 1
110000100 => 1
110000101 => 1
110000110 => 1
110000111 => 2
110001000 => 1
110001001 => 1
110001010 => 1
110001011 => 2
110001100 => 3
110001101 => 2
110001110 => 2
110001111 => 2
110010000 => 1
110010001 => 1
110010010 => 1
110010011 => 2
110010100 => 1
110010101 => 0
110010110 => 0
110010111 => 0
110011000 => 1
110011001 => 0
110011010 => 0
110011011 => 0
110011100 => 0
110011101 => 0
110011110 => 0
110011111 => 0
110100000 => 1
110100001 => 1
110100010 => 1
110100011 => 2
110100100 => 1
110100101 => 0
110100110 => 0
110100111 => 0
110101000 => 1
110101001 => 0
110101010 => 0
110101011 => 0
110101100 => 0
110101101 => 0
110101110 => 0
110101111 => 0
110110000 => 1
110110001 => 0
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 => 1
111000011 => 2
111000100 => 1
111000101 => 0
111000110 => 0
111000111 => 0
111001000 => 1
111001001 => 0
111001010 => 0
111001011 => 0
111001100 => 0
111001101 => 0
111001110 => 0
111001111 => 0
111010000 => 1
111010001 => 0
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 => 0
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: Jun 21, 2021 at 14:55 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Jun 21, 2021 at 14:55 by Martin Rubey