***************************************************************************** * 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: St000847 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The number of standard Young tableaux whose descent set is the binary word. A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation. For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(w): D = [i+1 for i in range(len(w)) if w[i]==1] return len([T for T in StandardTableaux(len(w)+1) if T.standard_descents() == D]) ----------------------------------------------------------------------------- Statistic values: 0 => 1 1 => 1 00 => 1 01 => 1 10 => 1 11 => 1 000 => 1 001 => 1 010 => 2 011 => 1 100 => 1 101 => 2 110 => 1 111 => 1 0000 => 1 0001 => 1 0010 => 2 0011 => 1 0100 => 2 0101 => 3 0110 => 2 0111 => 1 1000 => 1 1001 => 2 1010 => 3 1011 => 2 1100 => 1 1101 => 2 1110 => 1 1111 => 1 00000 => 1 00001 => 1 00010 => 2 00011 => 1 00100 => 3 00101 => 3 00110 => 2 00111 => 1 01000 => 2 01001 => 4 01010 => 6 01011 => 3 01100 => 2 01101 => 4 01110 => 2 01111 => 1 10000 => 1 10001 => 2 10010 => 4 10011 => 2 10100 => 3 10101 => 6 10110 => 4 10111 => 2 11000 => 1 11001 => 2 11010 => 3 11011 => 3 11100 => 1 11101 => 2 11110 => 1 11111 => 1 000000 => 1 000001 => 1 000010 => 2 000011 => 1 000100 => 3 000101 => 3 000110 => 2 000111 => 1 001000 => 3 001001 => 5 001010 => 7 001011 => 3 001100 => 3 001101 => 4 001110 => 2 001111 => 1 010000 => 2 010001 => 4 010010 => 8 010011 => 4 010100 => 7 010101 => 11 010110 => 7 010111 => 3 011000 => 2 011001 => 5 011010 => 7 011011 => 5 011100 => 2 011101 => 4 011110 => 2 011111 => 1 100000 => 1 100001 => 2 100010 => 4 100011 => 2 100100 => 5 100101 => 7 100110 => 5 100111 => 2 101000 => 3 101001 => 7 101010 => 11 101011 => 7 101100 => 4 101101 => 8 101110 => 4 101111 => 2 110000 => 1 110001 => 2 110010 => 4 110011 => 3 110100 => 3 110101 => 7 110110 => 5 110111 => 3 111000 => 1 111001 => 2 111010 => 3 111011 => 3 111100 => 1 111101 => 2 111110 => 1 111111 => 1 0000000 => 1 0000001 => 1 0000010 => 2 0000011 => 1 0000100 => 3 0000101 => 3 0000110 => 2 0000111 => 1 0001000 => 4 0001001 => 5 0001010 => 7 0001011 => 3 0001100 => 3 0001101 => 4 0001110 => 2 0001111 => 1 0010000 => 3 0010001 => 6 0010010 => 11 0010011 => 5 0010100 => 10 0010101 => 13 0010110 => 8 0010111 => 3 0011000 => 3 0011001 => 7 0011010 => 9 0011011 => 5 0011100 => 3 0011101 => 4 0011110 => 2 0011111 => 1 0100000 => 2 0100001 => 4 0100010 => 9 0100011 => 4 0100100 => 11 0100101 => 15 0100110 => 10 0100111 => 4 0101000 => 7 0101001 => 16 0101010 => 24 0101011 => 13 0101100 => 9 0101101 => 15 0101110 => 7 0101111 => 3 0110000 => 2 0110001 => 5 0110010 => 10 0110011 => 7 0110100 => 8 0110101 => 16 0110110 => 11 0110111 => 5 0111000 => 2 0111001 => 5 0111010 => 7 0111011 => 6 0111100 => 2 0111101 => 4 0111110 => 2 0111111 => 1 1000000 => 1 1000001 => 2 1000010 => 4 1000011 => 2 1000100 => 6 1000101 => 7 1000110 => 5 1000111 => 2 1001000 => 5 1001001 => 11 1001010 => 16 1001011 => 8 1001100 => 7 1001101 => 10 1001110 => 5 1001111 => 2 1010000 => 3 1010001 => 7 1010010 => 15 1010011 => 9 1010100 => 13 1010101 => 24 1010110 => 16 1010111 => 7 1011000 => 4 1011001 => 10 1011010 => 15 1011011 => 11 1011100 => 4 1011101 => 9 1011110 => 4 1011111 => 2 1100000 => 1 1100001 => 2 1100010 => 4 1100011 => 3 1100100 => 5 1100101 => 9 1100110 => 7 1100111 => 3 1101000 => 3 1101001 => 8 1101010 => 13 1101011 => 10 1101100 => 5 1101101 => 11 1101110 => 6 1101111 => 3 1110000 => 1 1110001 => 2 1110010 => 4 1110011 => 3 1110100 => 3 1110101 => 7 1110110 => 5 1110111 => 4 1111000 => 1 1111001 => 2 1111010 => 3 1111011 => 3 1111100 => 1 1111101 => 2 1111110 => 1 1111111 => 1 00000000 => 1 00000001 => 1 00000010 => 2 00000011 => 1 00000100 => 3 00000101 => 3 00000110 => 2 00000111 => 1 00001000 => 4 00001001 => 5 00001010 => 7 00001011 => 3 00001100 => 3 00001101 => 4 00001110 => 2 00001111 => 1 00010000 => 4 00010001 => 7 00010010 => 12 00010011 => 5 00010100 => 11 00010101 => 13 00010110 => 8 00010111 => 3 00011000 => 4 00011001 => 7 00011010 => 9 00011011 => 5 00011100 => 3 00011101 => 4 00011110 => 2 00011111 => 1 00100000 => 3 00100001 => 6 00100010 => 13 00100011 => 6 00100100 => 16 00100101 => 20 00100110 => 13 00100111 => 5 00101000 => 11 00101001 => 22 00101010 => 31 00101011 => 15 00101100 => 12 00101101 => 17 00101110 => 8 00101111 => 3 00110000 => 3 00110001 => 8 00110010 => 15 00110011 => 9 00110100 => 12 00110101 => 20 00110110 => 13 00110111 => 5 00111000 => 3 00111001 => 7 00111010 => 9 00111011 => 6 00111100 => 3 00111101 => 4 00111110 => 2 00111111 => 1 01000000 => 2 01000001 => 4 01000010 => 9 01000011 => 4 01000100 => 13 01000101 => 16 01000110 => 11 01000111 => 4 01001000 => 12 01001001 => 24 01001010 => 35 01001011 => 17 01001100 => 15 01001101 => 21 01001110 => 10 01001111 => 4 01010000 => 7 01010001 => 17 01010010 => 35 01010011 => 20 01010100 => 31 01010101 => 51 01010110 => 33 01010111 => 13 01011000 => 9 01011001 => 23 01011010 => 32 01011011 => 20 01011100 => 9 01011101 => 16 01011110 => 7 01011111 => 3 01100000 => 2 01100001 => 5 01100010 => 11 01100011 => 7 01100100 => 13 01100101 => 23 01100110 => 17 01100111 => 7 01101000 => 8 01101001 => 21 01101010 => 33 01101011 => 22 01101100 => 13 01101101 => 24 01101110 => 12 01101111 => 5 01110000 => 2 01110001 => 5 01110010 => 10 01110011 => 8 01110100 => 8 01110101 => 17 01110110 => 12 01110111 => 7 01111000 => 2 01111001 => 5 01111010 => 7 01111011 => 6 01111100 => 2 01111101 => 4 01111110 => 2 01111111 => 1 10000000 => 1 10000001 => 2 10000010 => 4 10000011 => 2 10000100 => 6 10000101 => 7 10000110 => 5 10000111 => 2 10001000 => 7 10001001 => 12 10001010 => 17 10001011 => 8 10001100 => 8 10001101 => 10 10001110 => 5 10001111 => 2 10010000 => 5 10010001 => 12 10010010 => 24 10010011 => 13 10010100 => 22 10010101 => 33 10010110 => 21 10010111 => 8 10011000 => 7 10011001 => 17 10011010 => 23 10011011 => 13 10011100 => 7 10011101 => 11 10011110 => 5 10011111 => 2 10100000 => 3 10100001 => 7 10100010 => 16 10100011 => 9 10100100 => 20 10100101 => 32 10100110 => 23 10100111 => 9 10101000 => 13 10101001 => 33 10101010 => 51 10101011 => 31 10101100 => 20 10101101 => 35 10101110 => 17 10101111 => 7 10110000 => 4 10110001 => 10 10110010 => 21 10110011 => 15 10110100 => 17 10110101 => 35 10110110 => 24 10110111 => 12 10111000 => 4 10111001 => 11 10111010 => 16 10111011 => 13 10111100 => 4 10111101 => 9 10111110 => 4 10111111 => 2 11000000 => 1 11000001 => 2 11000010 => 4 11000011 => 3 11000100 => 6 11000101 => 9 11000110 => 7 11000111 => 3 11001000 => 5 11001001 => 13 11001010 => 20 11001011 => 12 11001100 => 9 11001101 => 15 11001110 => 8 11001111 => 3 11010000 => 3 11010001 => 8 11010010 => 17 11010011 => 12 11010100 => 15 11010101 => 31 11010110 => 22 11010111 => 11 11011000 => 5 11011001 => 13 11011010 => 20 11011011 => 16 11011100 => 6 11011101 => 13 11011110 => 6 11011111 => 3 11100000 => 1 11100001 => 2 11100010 => 4 11100011 => 3 11100100 => 5 11100101 => 9 11100110 => 7 11100111 => 4 11101000 => 3 11101001 => 8 11101010 => 13 11101011 => 11 11101100 => 5 11101101 => 12 11101110 => 7 11101111 => 4 11110000 => 1 11110001 => 2 11110010 => 4 11110011 => 3 11110100 => 3 11110101 => 7 11110110 => 5 11110111 => 4 11111000 => 1 11111001 => 2 11111010 => 3 11111011 => 3 11111100 => 1 11111101 => 2 11111110 => 1 11111111 => 1 000000000 => 1 000000001 => 1 000000010 => 2 000000011 => 1 000000100 => 3 000000101 => 3 000000110 => 2 000000111 => 1 000001000 => 4 000001001 => 5 000001010 => 7 000001011 => 3 000001100 => 3 000001101 => 4 000001110 => 2 000001111 => 1 000010000 => 5 000010001 => 7 000010010 => 12 000010011 => 5 000010100 => 11 000010101 => 13 000010110 => 8 000010111 => 3 000011000 => 4 000011001 => 7 000011010 => 9 000011011 => 5 000011100 => 3 000011101 => 4 000011110 => 2 000011111 => 1 000100000 => 4 000100001 => 8 000100010 => 16 000100011 => 7 000100100 => 19 000100101 => 22 000100110 => 14 000100111 => 5 000101000 => 14 000101001 => 24 000101010 => 33 000101011 => 15 000101100 => 13 000101101 => 17 000101110 => 8 000101111 => 3 000110000 => 4 000110001 => 10 000110010 => 17 000110011 => 9 000110100 => 14 000110101 => 20 000110110 => 13 000110111 => 5 000111000 => 4 000111001 => 7 000111010 => 9 000111011 => 6 000111100 => 3 000111101 => 4 000111110 => 2 000111111 => 1 001000000 => 3 001000001 => 6 001000010 => 14 001000011 => 6 001000100 => 20 001000101 => 24 001000110 => 16 001000111 => 6 001001000 => 19 001001001 => 35 001001010 => 50 001001011 => 23 001001100 => 21 001001101 => 28 001001110 => 13 001001111 => 5 001010000 => 11 001010001 => 27 001010010 => 52 001010011 => 27 001010100 => 46 001010101 => 67 001010110 => 42 001010111 => 15 001011000 => 14 001011001 => 31 001011010 => 41 001011011 => 23 001011100 => 12 001011101 => 18 001011110 => 8 001011111 => 3 001100000 => 3 001100001 => 8 001100010 => 18 001100011 => 11 001100100 => 21 001100101 => 34 001100110 => 24 001100111 => 9 001101000 => 13 001101001 => 32 001101010 => 47 001101011 => 27 001101100 => 19 001101101 => 29 001101110 => 14 001101111 => 5 001110000 => 3 001110001 => 8 001110010 => 15 001110011 => 11 001110100 => 12 001110101 => 22 001110110 => 15 001110111 => 7 001111000 => 3 001111001 => 7 001111010 => 9 001111011 => 6 001111100 => 3 001111101 => 4 001111110 => 2 001111111 => 1 010000000 => 2 010000001 => 4 010000010 => 9 010000011 => 4 010000100 => 14 010000101 => 16 010000110 => 11 010000111 => 4 010001000 => 16 010001001 => 28 010001010 => 40 010001011 => 18 010001100 => 18 010001101 => 23 010001110 => 11 010001111 => 4 010010000 => 12 010010001 => 29 010010010 => 57 010010011 => 29 010010100 => 52 010010101 => 75 010010110 => 47 010010111 => 17 010011000 => 17 010011001 => 38 010011010 => 51 010011011 => 28 010011100 => 15 010011101 => 23 010011110 => 10 010011111 => 4 010100000 => 7 010100001 => 17 010100010 => 40 010100011 => 22 010100100 => 50 010100101 => 77 010100110 => 54 010100111 => 20 010101000 => 33 010101001 => 80 010101010 => 120 010101011 => 67 010101100 => 47 010101101 => 75 010101110 => 35 010101111 => 13 010110000 => 9 010110001 => 25 010110010 => 51 010110011 => 34 010110100 => 41 010110101 => 77 010110110 => 51 010110111 => 22 010111000 => 9 010111001 => 25 010111010 => 34 010111011 => 24 010111100 => 9 010111101 => 16 010111110 => 7 010111111 => 3 011000000 => 2 011000001 => 5 011000010 => 11 011000011 => 7 011000100 => 16 011000101 => 25 011000110 => 19 011000111 => 7 011001000 => 14 011001001 => 35 011001010 => 54 011001011 => 31 011001100 => 24 011001101 => 38 011001110 => 19 011001111 => 7 011010000 => 8 011010001 => 22 011010010 => 47 011010011 => 32 011010100 => 42 011010101 => 80 011010110 => 55 011010111 => 24 011011000 => 13 011011001 => 35 011011010 => 51 011011011 => 35 011011100 => 15 011011101 => 28 011011110 => 12 011011111 => 5 011100000 => 2 011100001 => 5 011100010 => 11 011100011 => 8 011100100 => 13 011100101 => 25 011100110 => 19 011100111 => 10 011101000 => 8 011101001 => 22 011101010 => 35 011101011 => 27 011101100 => 14 011101101 => 29 011101110 => 16 011101111 => 7 011110000 => 2 011110001 => 5 011110010 => 10 011110011 => 8 011110100 => 8 011110101 => 17 011110110 => 12 011110111 => 8 011111000 => 2 011111001 => 5 011111010 => 7 011111011 => 6 011111100 => 2 011111101 => 4 011111110 => 2 011111111 => 1 100000000 => 1 100000001 => 2 100000010 => 4 100000011 => 2 100000100 => 6 100000101 => 7 100000110 => 5 100000111 => 2 100001000 => 8 100001001 => 12 100001010 => 17 100001011 => 8 100001100 => 8 100001101 => 10 100001110 => 5 100001111 => 2 100010000 => 7 100010001 => 16 100010010 => 29 100010011 => 14 100010100 => 27 100010101 => 35 100010110 => 22 100010111 => 8 100011000 => 10 100011001 => 19 100011010 => 25 100011011 => 13 100011100 => 8 100011101 => 11 100011110 => 5 100011111 => 2 100100000 => 5 100100001 => 12 100100010 => 28 100100011 => 15 100100100 => 35 100100101 => 51 100100110 => 35 100100111 => 13 100101000 => 24 100101001 => 55 100101010 => 80 100101011 => 42 100101100 => 32 100101101 => 47 100101110 => 22 100101111 => 8 100110000 => 7 100110001 => 19 100110010 => 38 100110011 => 24 100110100 => 31 100110101 => 54 100110110 => 35 100110111 => 14 100111000 => 7 100111001 => 19 100111010 => 25 100111011 => 16 100111100 => 7 100111101 => 11 100111110 => 5 100111111 => 2 101000000 => 3 101000001 => 7 101000010 => 16 101000011 => 9 101000100 => 24 101000101 => 34 101000110 => 25 101000111 => 9 101001000 => 22 101001001 => 51 101001010 => 77 101001011 => 41 101001100 => 34 101001101 => 51 101001110 => 25 101001111 => 9 101010000 => 13 101010001 => 35 101010010 => 75 101010011 => 47 101010100 => 67 101010101 => 120 101010110 => 80 101010111 => 33 101011000 => 20 101011001 => 54 101011010 => 77 101011011 => 50 101011100 => 22 101011101 => 40 101011110 => 17 101011111 => 7 101100000 => 4 101100001 => 10 101100010 => 23 101100011 => 15 101100100 => 28 101100101 => 51 101100110 => 38 101100111 => 17 101101000 => 17 101101001 => 47 101101010 => 75 101101011 => 52 101101100 => 29 101101101 => 57 101101110 => 29 101101111 => 12 101110000 => 4 101110001 => 11 101110010 => 23 101110011 => 18 101110100 => 18 101110101 => 40 101110110 => 28 101110111 => 16 101111000 => 4 101111001 => 11 101111010 => 16 101111011 => 14 101111100 => 4 101111101 => 9 101111110 => 4 101111111 => 2 110000000 => 1 110000001 => 2 110000010 => 4 110000011 => 3 110000100 => 6 110000101 => 9 110000110 => 7 110000111 => 3 110001000 => 7 110001001 => 15 110001010 => 22 110001011 => 12 110001100 => 11 110001101 => 15 110001110 => 8 110001111 => 3 110010000 => 5 110010001 => 14 110010010 => 29 110010011 => 19 110010100 => 27 110010101 => 47 110010110 => 32 110010111 => 13 110011000 => 9 110011001 => 24 110011010 => 34 110011011 => 21 110011100 => 11 110011101 => 18 110011110 => 8 110011111 => 3 110100000 => 3 110100001 => 8 110100010 => 18 110100011 => 12 110100100 => 23 110100101 => 41 110100110 => 31 110100111 => 14 110101000 => 15 110101001 => 42 110101010 => 67 110101011 => 46 110101100 => 27 110101101 => 52 110101110 => 27 110101111 => 11 110110000 => 5 110110001 => 13 110110010 => 28 110110011 => 21 110110100 => 23 110110101 => 50 110110110 => 35 110110111 => 19 110111000 => 6 110111001 => 16 110111010 => 24 110111011 => 20 110111100 => 6 110111101 => 14 110111110 => 6 110111111 => 3 111000000 => 1 111000001 => 2 111000010 => 4 111000011 => 3 111000100 => 6 111000101 => 9 111000110 => 7 111000111 => 4 111001000 => 5 111001001 => 13 111001010 => 20 111001011 => 14 111001100 => 9 111001101 => 17 111001110 => 10 111001111 => 4 111010000 => 3 111010001 => 8 111010010 => 17 111010011 => 13 111010100 => 15 111010101 => 33 111010110 => 24 111010111 => 14 111011000 => 5 111011001 => 14 111011010 => 22 111011011 => 19 111011100 => 7 111011101 => 16 111011110 => 8 111011111 => 4 111100000 => 1 111100001 => 2 111100010 => 4 111100011 => 3 111100100 => 5 111100101 => 9 111100110 => 7 111100111 => 4 111101000 => 3 111101001 => 8 111101010 => 13 111101011 => 11 111101100 => 5 111101101 => 12 111101110 => 7 111101111 => 5 111110000 => 1 111110001 => 2 111110010 => 4 111110011 => 3 111110100 => 3 111110101 => 7 111110110 => 5 111110111 => 4 111111000 => 1 111111001 => 2 111111010 => 3 111111011 => 3 111111100 => 1 111111101 => 2 111111110 => 1 111111111 => 1 0000000001 => 1 0010000001 => 6 0010101101 => 96 0010101011 => 87 0010100111 => 27 0010011101 => 30 0010011011 => 37 0010010111 => 23 0010001111 => 6 0001110101 => 22 0001110011 => 11 0001101101 => 29 0001101011 => 27 0001100111 => 9 0001011101 => 18 0001011011 => 23 0001010111 => 15 0001001111 => 5 0000111101 => 4 0000111011 => 6 0000110111 => 5 0000101111 => 3 0000011111 => 1 0000000000 => 1 0000000110 => 2 0000011000 => 4 0000011110 => 2 0000010010 => 12 0001100000 => 4 0001100110 => 26 0001111000 => 4 0001111110 => 2 0001110010 => 17 0001001000 => 24 0001001110 => 14 0001000010 => 18 0001011010 => 43 0010101010 => 169 0010101000 => 53 0010101100 => 67 0010100100 => 78 0010010100 => 78 0001010100 => 53 0010101110 => 44 0010100000 => 11 0010000100 => 22 0000010100 => 11 0000000010 => 2 0000001110 => 2 0000111110 => 2 0000000100 => 3 0000001010 => 7 0000010110 => 8 0000101110 => 8 0001011110 => 8 0001101110 => 14 0001110110 => 15 0001111010 => 9 0001111100 => 3 0000000101 => 3 0000001101 => 4 0000011101 => 4 0001111101 => 4 0000001001 => 5 0000010101 => 13 0000101101 => 17 0010110001 => 39 0001100001 => 11 0001000001 => 8 0000111111 => 1 0001110100 => 14 0001101100 => 21 0010011100 => 21 0001011100 => 13 0000111100 => 3 0001101010 => 51 0010011010 => 71 0000111010 => 9 0010100110 => 77 0010010110 => 66 0001010110 => 44 0000110110 => 13 0010001110 => 16 0001101001 => 36 0000110001 => 10 0000010001 => 7 0000001100 => 3 0000011001 => 7 0000110011 => 9 0000110100 => 14 0000000011 => 1 0000011011 => 5 0001101111 => 5 0000110101 => 20 0010000010 => 14 0001000011 => 8 0010011110 => 13 0001010000 => 15 0001000100 => 25 0001010101 => 71 0001001011 => 25 0001100011 => 13 0001110111 => 7 0010000000 => 3 0000000111 => 1 0000001111 => 1 0001111111 => 1 0000001011 => 3 0000010111 => 3 0001011111 => 3 0001111001 => 7 0000011010 => 9 0000101010 => 33 0000100101 => 22 0010010001 => 45 0010000101 => 25 0010010101 => 107 0010010011 => 42 0000101011 => 15 0001000111 => 7 0000100111 => 5 0010010010 => 86 0001001010 => 57 0001001001 => 41 0001000101 => 29 0000100011 => 7 0000100100 => 20 0000100010 => 17 0010110000 => 14 0000011100 => 3 0000001000 => 4 0000101100 => 13 0010101111 => 15 0000101000 => 15 0010001000 => 25 0001001100 => 24 0001000110 => 19 0001100100 => 26 0001100010 => 23 0000100110 => 14 0000110010 => 17 0010010000 => 20 0001010010 => 60 0001000000 => 4 0001011001 => 33 0001100101 => 38 0001110001 => 10 0010011001 => 54 0010100101 => 113 0010101001 => 117 0010011111 => 5 0000111001 => 7 0010001101 => 34 0001001101 => 30 0001010001 => 33 0001010011 => 29 0001111011 => 6 0010011000 => 26 0001011000 => 17 0000100000 => 5 0010001011 => 27 0010000111 => 6 0000010011 => 5 0010001100 => 27 0010000110 => 17 0000110000 => 5 0000100001 => 9 0010100011 => 34 0010100001 => 28 0000111000 => 4 0001101000 => 17 0000010000 => 5 0000101001 => 24 0010100010 => 64 0010001001 => 43 0010000011 => 6 0010001010 => 61 0001110000 => 4 ----------------------------------------------------------------------------- Created: Jun 15, 2017 at 08:07 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jun 15, 2017 at 08:07 by Martin Rubey