***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * 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: St001313 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$. See [[St001312]] for this statistic on compositions treated as bounce paths. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def lattice_paths_above_boundary(B): B = list(B) if sum(B) == 0: return [B] else: paths = [ B[:1] + p for p in lattice_paths_above_boundary(B[1:]) ] if B[0] == 0: i = B.index(1) B = B[:i] + B[i+1:] paths.extend( [ [1] + p for p in lattice_paths_above_boundary(B) ] ) return paths def statistic(w): return len(lattice_paths_above_boundary(w)) ----------------------------------------------------------------------------- Statistic values: 0 => 1 1 => 1 00 => 1 01 => 2 10 => 1 11 => 1 000 => 1 001 => 3 010 => 2 011 => 3 100 => 1 101 => 2 110 => 1 111 => 1 0000 => 1 0001 => 4 0010 => 3 0011 => 6 0100 => 2 0101 => 5 0110 => 3 0111 => 4 1000 => 1 1001 => 3 1010 => 2 1011 => 3 1100 => 1 1101 => 2 1110 => 1 1111 => 1 00000 => 1 00001 => 5 00010 => 4 00011 => 10 00100 => 3 00101 => 9 00110 => 6 00111 => 10 01000 => 2 01001 => 7 01010 => 5 01011 => 9 01100 => 3 01101 => 7 01110 => 4 01111 => 5 10000 => 1 10001 => 4 10010 => 3 10011 => 6 10100 => 2 10101 => 5 10110 => 3 10111 => 4 11000 => 1 11001 => 3 11010 => 2 11011 => 3 11100 => 1 11101 => 2 11110 => 1 11111 => 1 000000 => 1 000001 => 6 000010 => 5 000011 => 15 000100 => 4 000101 => 14 000110 => 10 000111 => 20 001000 => 3 001001 => 12 001010 => 9 001011 => 19 001100 => 6 001101 => 16 001110 => 10 001111 => 15 010000 => 2 010001 => 9 010010 => 7 010011 => 16 010100 => 5 010101 => 14 010110 => 9 010111 => 14 011000 => 3 011001 => 10 011010 => 7 011011 => 12 011100 => 4 011101 => 9 011110 => 5 011111 => 6 100000 => 1 100001 => 5 100010 => 4 100011 => 10 100100 => 3 100101 => 9 100110 => 6 100111 => 10 101000 => 2 101001 => 7 101010 => 5 101011 => 9 101100 => 3 101101 => 7 101110 => 4 101111 => 5 110000 => 1 110001 => 4 110010 => 3 110011 => 6 110100 => 2 110101 => 5 110110 => 3 110111 => 4 111000 => 1 111001 => 3 111010 => 2 111011 => 3 111100 => 1 111101 => 2 111110 => 1 111111 => 1 0000000 => 1 0000001 => 7 0000010 => 6 0000011 => 21 0000100 => 5 0000101 => 20 0000110 => 15 0000111 => 35 0001000 => 4 0001001 => 18 0001010 => 14 0001011 => 34 0001100 => 10 0001101 => 30 0001110 => 20 0001111 => 35 0010000 => 3 0010001 => 15 0010010 => 12 0010011 => 31 0010100 => 9 0010101 => 28 0010110 => 19 0010111 => 34 0011000 => 6 0011001 => 22 0011010 => 16 0011011 => 31 0011100 => 10 0011101 => 25 0011110 => 15 0011111 => 21 0100000 => 2 0100001 => 11 0100010 => 9 0100011 => 25 0100100 => 7 0100101 => 23 0100110 => 16 0100111 => 30 0101000 => 5 0101001 => 19 0101010 => 14 0101011 => 28 0101100 => 9 0101101 => 23 0101110 => 14 0101111 => 20 0110000 => 3 0110001 => 13 0110010 => 10 0110011 => 22 0110100 => 7 0110101 => 19 0110110 => 12 0110111 => 18 0111000 => 4 0111001 => 13 0111010 => 9 0111011 => 15 0111100 => 5 0111101 => 11 0111110 => 6 0111111 => 7 1000000 => 1 1000001 => 6 1000010 => 5 1000011 => 15 1000100 => 4 1000101 => 14 1000110 => 10 1000111 => 20 1001000 => 3 1001001 => 12 1001010 => 9 1001011 => 19 1001100 => 6 1001101 => 16 1001110 => 10 1001111 => 15 1010000 => 2 1010001 => 9 1010010 => 7 1010011 => 16 1010100 => 5 1010101 => 14 1010110 => 9 1010111 => 14 1011000 => 3 1011001 => 10 1011010 => 7 1011011 => 12 1011100 => 4 1011101 => 9 1011110 => 5 1011111 => 6 1100000 => 1 1100001 => 5 1100010 => 4 1100011 => 10 1100100 => 3 1100101 => 9 1100110 => 6 1100111 => 10 1101000 => 2 1101001 => 7 1101010 => 5 1101011 => 9 1101100 => 3 1101101 => 7 1101110 => 4 1101111 => 5 1110000 => 1 1110001 => 4 1110010 => 3 1110011 => 6 1110100 => 2 1110101 => 5 1110110 => 3 1110111 => 4 1111000 => 1 1111001 => 3 1111010 => 2 1111011 => 3 1111100 => 1 1111101 => 2 1111110 => 1 1111111 => 1 00000000 => 1 00000001 => 8 00000010 => 7 00000011 => 28 00000100 => 6 00000101 => 27 00000110 => 21 00000111 => 56 00001000 => 5 00001001 => 25 00001010 => 20 00001011 => 55 00001100 => 15 00001101 => 50 00001110 => 35 00001111 => 70 00010000 => 4 00010001 => 22 00010010 => 18 00010011 => 52 00010100 => 14 00010101 => 48 00010110 => 34 00010111 => 69 00011000 => 10 00011001 => 40 00011010 => 30 00011011 => 65 00011100 => 20 00011101 => 55 00011110 => 35 00011111 => 56 00100000 => 3 00100001 => 18 00100010 => 15 00100011 => 46 00100100 => 12 00100101 => 43 00100110 => 31 00100111 => 65 00101000 => 9 00101001 => 37 00101010 => 28 00101011 => 62 00101100 => 19 00101101 => 53 00101110 => 34 00101111 => 55 00110000 => 6 00110001 => 28 00110010 => 22 00110011 => 53 00110100 => 16 00110101 => 47 00110110 => 31 00110111 => 52 00111000 => 10 00111001 => 35 00111010 => 25 00111011 => 46 00111100 => 15 00111101 => 36 00111110 => 21 00111111 => 28 01000000 => 2 01000001 => 13 01000010 => 11 01000011 => 36 01000100 => 9 01000101 => 34 01000110 => 25 01000111 => 55 01001000 => 7 01001001 => 30 01001010 => 23 01001011 => 53 01001100 => 16 01001101 => 46 01001110 => 30 01001111 => 50 01010000 => 5 01010001 => 24 01010010 => 19 01010011 => 47 01010100 => 14 01010101 => 42 01010110 => 28 01010111 => 48 01011000 => 9 01011001 => 32 01011010 => 23 01011011 => 43 01011100 => 14 01011101 => 34 01011110 => 20 01011111 => 27 01100000 => 3 01100001 => 16 01100010 => 13 01100011 => 35 01100100 => 10 01100101 => 32 01100110 => 22 01100111 => 40 01101000 => 7 01101001 => 26 01101010 => 19 01101011 => 37 01101100 => 12 01101101 => 30 01101110 => 18 01101111 => 25 01110000 => 4 01110001 => 17 01110010 => 13 01110011 => 28 01110100 => 9 01110101 => 24 01110110 => 15 01110111 => 22 01111000 => 5 01111001 => 16 01111010 => 11 01111011 => 18 01111100 => 6 01111101 => 13 01111110 => 7 01111111 => 8 10000000 => 1 10000001 => 7 10000010 => 6 10000011 => 21 10000100 => 5 10000101 => 20 10000110 => 15 10000111 => 35 10001000 => 4 10001001 => 18 10001010 => 14 10001011 => 34 10001100 => 10 10001101 => 30 10001110 => 20 10001111 => 35 10010000 => 3 10010001 => 15 10010010 => 12 10010011 => 31 10010100 => 9 10010101 => 28 10010110 => 19 10010111 => 34 10011000 => 6 10011001 => 22 10011010 => 16 10011011 => 31 10011100 => 10 10011101 => 25 10011110 => 15 10011111 => 21 10100000 => 2 10100001 => 11 10100010 => 9 10100011 => 25 10100100 => 7 10100101 => 23 10100110 => 16 10100111 => 30 10101000 => 5 10101001 => 19 10101010 => 14 10101011 => 28 10101100 => 9 10101101 => 23 10101110 => 14 10101111 => 20 10110000 => 3 10110001 => 13 10110010 => 10 10110011 => 22 10110100 => 7 10110101 => 19 10110110 => 12 10110111 => 18 10111000 => 4 10111001 => 13 10111010 => 9 10111011 => 15 10111100 => 5 10111101 => 11 10111110 => 6 10111111 => 7 11000000 => 1 11000001 => 6 11000010 => 5 11000011 => 15 11000100 => 4 11000101 => 14 11000110 => 10 11000111 => 20 11001000 => 3 11001001 => 12 11001010 => 9 11001011 => 19 11001100 => 6 11001101 => 16 11001110 => 10 11001111 => 15 11010000 => 2 11010001 => 9 11010010 => 7 11010011 => 16 11010100 => 5 11010101 => 14 11010110 => 9 11010111 => 14 11011000 => 3 11011001 => 10 11011010 => 7 11011011 => 12 11011100 => 4 11011101 => 9 11011110 => 5 11011111 => 6 11100000 => 1 11100001 => 5 11100010 => 4 11100011 => 10 11100100 => 3 11100101 => 9 11100110 => 6 11100111 => 10 11101000 => 2 11101001 => 7 11101010 => 5 11101011 => 9 11101100 => 3 11101101 => 7 11101110 => 4 11101111 => 5 11110000 => 1 11110001 => 4 11110010 => 3 11110011 => 6 11110100 => 2 11110101 => 5 11110110 => 3 11110111 => 4 11111000 => 1 11111001 => 3 11111010 => 2 11111011 => 3 11111100 => 1 11111101 => 2 11111110 => 1 11111111 => 1 000000000 => 1 000000001 => 9 000000010 => 8 000000011 => 36 000000100 => 7 000000101 => 35 000000110 => 28 000000111 => 84 000001000 => 6 000001001 => 33 000001010 => 27 000001011 => 83 000001100 => 21 000001101 => 77 000001110 => 56 000001111 => 126 000010000 => 5 000010001 => 30 000010010 => 25 000010011 => 80 000010100 => 20 000010101 => 75 000010110 => 55 000010111 => 125 000011000 => 15 000011001 => 65 000011010 => 50 000011011 => 120 000011100 => 35 000011101 => 105 000011110 => 70 000011111 => 126 000100000 => 4 000100001 => 26 000100010 => 22 000100011 => 74 000100100 => 18 000100101 => 70 000100110 => 52 000100111 => 121 000101000 => 14 000101001 => 62 000101010 => 48 000101011 => 117 000101100 => 34 000101101 => 103 000101110 => 69 000101111 => 125 000110000 => 10 000110001 => 50 000110010 => 40 000110011 => 105 000110100 => 30 000110101 => 95 000110110 => 65 000110111 => 121 000111000 => 20 000111001 => 75 000111010 => 55 000111011 => 111 000111100 => 35 000111101 => 91 000111110 => 56 000111111 => 84 001000000 => 3 001000001 => 21 001000010 => 18 001000011 => 64 001000100 => 15 001000101 => 61 001000110 => 46 001000111 => 111 001001000 => 12 001001001 => 55 001001010 => 43 001001011 => 108 001001100 => 31 001001101 => 96 001001110 => 65 001001111 => 120 001010000 => 9 001010001 => 46 001010010 => 37 001010011 => 99 001010100 => 28 001010101 => 90 001010110 => 62 001010111 => 117 001011000 => 19 001011001 => 72 001011010 => 53 001011011 => 108 001011100 => 34 001011101 => 89 001011110 => 55 001011111 => 83 001100000 => 6 001100001 => 34 001100010 => 28 001100011 => 81 001100100 => 22 001100101 => 75 001100110 => 53 001100111 => 105 001101000 => 16 001101001 => 63 001101010 => 47 001101011 => 99 001101100 => 31 001101101 => 83 001101110 => 52 001101111 => 80 001110000 => 10 001110001 => 45 001110010 => 35 001110011 => 81 001110100 => 25 001110101 => 71 001110110 => 46 001110111 => 74 001111000 => 15 001111001 => 51 001111010 => 36 001111011 => 64 001111100 => 21 001111101 => 49 001111110 => 28 001111111 => 36 010000000 => 2 010000001 => 15 010000010 => 13 010000011 => 49 010000100 => 11 010000101 => 47 010000110 => 36 010000111 => 91 010001000 => 9 010001001 => 43 010001010 => 34 010001011 => 89 010001100 => 25 010001101 => 80 010001110 => 55 010001111 => 105 010010000 => 7 010010001 => 37 010010010 => 30 010010011 => 83 010010100 => 23 010010101 => 76 010010110 => 53 010010111 => 103 010011000 => 16 010011001 => 62 010011010 => 46 010011011 => 96 010011100 => 30 010011101 => 80 010011110 => 50 010011111 => 77 010100000 => 5 010100001 => 29 010100010 => 24 010100011 => 71 010100100 => 19 010100101 => 66 010100110 => 47 010100111 => 95 010101000 => 14 010101001 => 56 010101010 => 42 010101011 => 90 010101100 => 28 010101101 => 76 010101110 => 48 010101111 => 75 010110000 => 9 010110001 => 41 010110010 => 32 010110011 => 75 010110100 => 23 010110101 => 66 010110110 => 43 010110111 => 70 010111000 => 14 010111001 => 48 010111010 => 34 010111011 => 61 010111100 => 20 010111101 => 47 010111110 => 27 010111111 => 35 011000000 => 3 011000001 => 19 011000010 => 16 011000011 => 51 011000100 => 13 011000101 => 48 011000110 => 35 011000111 => 75 011001000 => 10 011001001 => 42 011001010 => 32 011001011 => 72 011001100 => 22 011001101 => 62 011001110 => 40 011001111 => 65 011010000 => 7 011010001 => 33 011010010 => 26 011010011 => 63 011010100 => 19 011010101 => 56 011010110 => 37 011010111 => 62 011011000 => 12 011011001 => 42 011011010 => 30 011011011 => 55 011011100 => 18 011011101 => 43 011011110 => 25 011011111 => 33 011100000 => 4 011100001 => 21 011100010 => 17 011100011 => 45 011100100 => 13 011100101 => 41 011100110 => 28 011100111 => 50 011101000 => 9 011101001 => 33 011101010 => 24 011101011 => 46 011101100 => 15 011101101 => 37 011101110 => 22 011101111 => 30 011110000 => 5 011110001 => 21 011110010 => 16 011110011 => 34 011110100 => 11 011110101 => 29 011110110 => 18 011110111 => 26 011111000 => 6 011111001 => 19 011111010 => 13 011111011 => 21 011111100 => 7 011111101 => 15 011111110 => 8 011111111 => 9 100000000 => 1 100000001 => 8 100000010 => 7 100000011 => 28 100000100 => 6 100000101 => 27 100000110 => 21 100000111 => 56 100001000 => 5 100001001 => 25 100001010 => 20 100001011 => 55 100001100 => 15 100001101 => 50 100001110 => 35 100001111 => 70 100010000 => 4 100010001 => 22 100010010 => 18 100010011 => 52 100010100 => 14 100010101 => 48 100010110 => 34 100010111 => 69 100011000 => 10 100011001 => 40 100011010 => 30 100011011 => 65 100011100 => 20 100011101 => 55 100011110 => 35 100011111 => 56 100100000 => 3 100100001 => 18 100100010 => 15 100100011 => 46 100100100 => 12 100100101 => 43 100100110 => 31 100100111 => 65 100101000 => 9 100101001 => 37 100101010 => 28 100101011 => 62 100101100 => 19 100101101 => 53 100101110 => 34 100101111 => 55 100110000 => 6 100110001 => 28 100110010 => 22 100110011 => 53 100110100 => 16 100110101 => 47 100110110 => 31 100110111 => 52 100111000 => 10 100111001 => 35 100111010 => 25 100111011 => 46 100111100 => 15 100111101 => 36 100111110 => 21 100111111 => 28 101000000 => 2 101000001 => 13 101000010 => 11 101000011 => 36 101000100 => 9 101000101 => 34 101000110 => 25 101000111 => 55 101001000 => 7 101001001 => 30 101001010 => 23 101001011 => 53 101001100 => 16 101001101 => 46 101001110 => 30 101001111 => 50 101010000 => 5 101010001 => 24 101010010 => 19 101010011 => 47 101010100 => 14 101010101 => 42 101010110 => 28 101010111 => 48 101011000 => 9 101011001 => 32 101011010 => 23 101011011 => 43 101011100 => 14 101011101 => 34 101011110 => 20 101011111 => 27 101100000 => 3 101100001 => 16 101100010 => 13 101100011 => 35 101100100 => 10 101100101 => 32 101100110 => 22 101100111 => 40 101101000 => 7 101101001 => 26 101101010 => 19 101101011 => 37 101101100 => 12 101101101 => 30 101101110 => 18 101101111 => 25 101110000 => 4 101110001 => 17 101110010 => 13 101110011 => 28 101110100 => 9 101110101 => 24 101110110 => 15 101110111 => 22 101111000 => 5 101111001 => 16 101111010 => 11 101111011 => 18 101111100 => 6 101111101 => 13 101111110 => 7 101111111 => 8 110000000 => 1 110000001 => 7 110000010 => 6 110000011 => 21 110000100 => 5 110000101 => 20 110000110 => 15 110000111 => 35 110001000 => 4 110001001 => 18 110001010 => 14 110001011 => 34 110001100 => 10 110001101 => 30 110001110 => 20 110001111 => 35 110010000 => 3 110010001 => 15 110010010 => 12 110010011 => 31 110010100 => 9 110010101 => 28 110010110 => 19 110010111 => 34 110011000 => 6 110011001 => 22 110011010 => 16 110011011 => 31 110011100 => 10 110011101 => 25 110011110 => 15 110011111 => 21 110100000 => 2 110100001 => 11 110100010 => 9 110100011 => 25 110100100 => 7 110100101 => 23 110100110 => 16 110100111 => 30 110101000 => 5 110101001 => 19 110101010 => 14 110101011 => 28 110101100 => 9 110101101 => 23 110101110 => 14 110101111 => 20 110110000 => 3 110110001 => 13 110110010 => 10 110110011 => 22 110110100 => 7 110110101 => 19 110110110 => 12 110110111 => 18 110111000 => 4 110111001 => 13 110111010 => 9 110111011 => 15 110111100 => 5 110111101 => 11 110111110 => 6 110111111 => 7 111000000 => 1 111000001 => 6 111000010 => 5 111000011 => 15 111000100 => 4 111000101 => 14 111000110 => 10 111000111 => 20 111001000 => 3 111001001 => 12 111001010 => 9 111001011 => 19 111001100 => 6 111001101 => 16 111001110 => 10 111001111 => 15 111010000 => 2 111010001 => 9 111010010 => 7 111010011 => 16 111010100 => 5 111010101 => 14 111010110 => 9 111010111 => 14 111011000 => 3 111011001 => 10 111011010 => 7 111011011 => 12 111011100 => 4 111011101 => 9 111011110 => 5 111011111 => 6 111100000 => 1 111100001 => 5 111100010 => 4 111100011 => 10 111100100 => 3 111100101 => 9 111100110 => 6 111100111 => 10 111101000 => 2 111101001 => 7 111101010 => 5 111101011 => 9 111101100 => 3 111101101 => 7 111101110 => 4 111101111 => 5 111110000 => 1 111110001 => 4 111110010 => 3 111110011 => 6 111110100 => 2 111110101 => 5 111110110 => 3 111110111 => 4 111111000 => 1 111111001 => 3 111111010 => 2 111111011 => 3 111111100 => 1 111111101 => 2 111111110 => 1 111111111 => 1 ----------------------------------------------------------------------------- Created: Dec 12, 2018 at 14:54 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Dec 12, 2018 at 14:54 by Christian Stump