Identifier
Identifier
Values
0 => 1
1 => 1
00 => 1
01 => 2
10 => 2
11 => 1
000 => 1
001 => 3
010 => 5
011 => 3
100 => 3
101 => 5
110 => 3
111 => 1
0000 => 1
0001 => 4
0010 => 9
0011 => 6
0100 => 9
0101 => 16
0110 => 11
0111 => 4
1000 => 4
1001 => 11
1010 => 16
1011 => 9
1100 => 6
1101 => 9
1110 => 4
1111 => 1
00000 => 1
00001 => 5
00010 => 14
00011 => 10
00100 => 19
00101 => 35
00110 => 26
00111 => 10
01000 => 14
01001 => 40
01010 => 61
01011 => 35
01100 => 26
01101 => 40
01110 => 19
01111 => 5
10000 => 5
10001 => 19
10010 => 40
10011 => 26
10100 => 35
10101 => 61
10110 => 40
10111 => 14
11000 => 10
11001 => 26
11010 => 35
11011 => 19
11100 => 10
11101 => 14
11110 => 5
11111 => 1
000000 => 1
000001 => 6
000010 => 20
000011 => 15
000100 => 34
000101 => 64
000110 => 50
000111 => 20
001000 => 34
001001 => 99
001010 => 155
001011 => 90
001100 => 71
001101 => 111
001110 => 55
001111 => 15
010000 => 20
010001 => 78
010010 => 169
010011 => 111
010100 => 155
010101 => 272
010110 => 181
010111 => 64
011000 => 50
011001 => 132
011010 => 181
011011 => 99
011100 => 55
011101 => 78
011110 => 29
011111 => 6
100000 => 6
100001 => 29
100010 => 78
100011 => 55
100100 => 99
100101 => 181
100110 => 132
100111 => 50
101000 => 64
101001 => 181
101010 => 272
101011 => 155
101100 => 111
101101 => 169
101110 => 78
101111 => 20
110000 => 15
110001 => 55
110010 => 111
110011 => 71
110100 => 90
110101 => 155
110110 => 99
110111 => 34
111000 => 20
111001 => 50
111010 => 64
111011 => 34
111100 => 15
111101 => 20
111110 => 6
111111 => 1
0000000 => 1
0000001 => 7
0000010 => 27
0000011 => 21
0000100 => 55
0000101 => 105
0000110 => 85
0000111 => 35
0001000 => 69
0001001 => 203
0001010 => 323
0001011 => 189
0001100 => 155
0001101 => 245
0001110 => 125
0001111 => 35
0010000 => 55
0010001 => 217
0010010 => 477
0010011 => 315
0010100 => 449
0010101 => 791
0010110 => 531
0010111 => 189
0011000 => 155
0011001 => 413
0011010 => 573
0011011 => 315
0011100 => 181
0011101 => 259
0011110 => 99
0011111 => 21
0100000 => 27
0100001 => 133
0100010 => 365
0100011 => 259
0100100 => 477
0100101 => 875
0100110 => 643
0100111 => 245
0101000 => 323
0101001 => 917
0101010 => 1385
0101011 => 791
0101100 => 573
0101101 => 875
0101110 => 407
0101111 => 105
0110000 => 85
0110001 => 315
0110010 => 643
0110011 => 413
0110100 => 531
0110101 => 917
0110110 => 589
0110111 => 203
0111000 => 125
0111001 => 315
0111010 => 407
0111011 => 217
0111100 => 99
0111101 => 133
0111110 => 41
0111111 => 7
1000000 => 7
1000001 => 41
1000010 => 133
1000011 => 99
1000100 => 217
1000101 => 407
1000110 => 315
1000111 => 125
1001000 => 203
1001001 => 589
1001010 => 917
1001011 => 531
1001100 => 413
1001101 => 643
1001110 => 315
1001111 => 85
1010000 => 105
1010001 => 407
1010010 => 875
1010011 => 573
1010100 => 791
1010101 => 1385
1010110 => 917
1010111 => 323
1011000 => 245
1011001 => 643
1011010 => 875
1011011 => 477
1011100 => 259
1011101 => 365
1011110 => 133
1011111 => 27
1100000 => 21
1100001 => 99
1100010 => 259
1100011 => 181
1100100 => 315
1100101 => 573
1100110 => 413
1100111 => 155
1101000 => 189
1101001 => 531
1101010 => 791
1101011 => 449
1101100 => 315
1101101 => 477
1101110 => 217
1101111 => 55
1110000 => 35
1110001 => 125
1110010 => 245
1110011 => 155
1110100 => 189
1110101 => 323
1110110 => 203
1110111 => 69
1111000 => 35
1111001 => 85
1111010 => 105
1111011 => 55
1111100 => 21
1111101 => 27
1111110 => 7
1111111 => 1
click to show generating function       
Description
The number of permutations whose descent word is the given binary word.
This is the sizes of the preimages of the map Mp00109descent word.
Code
from collections import defaultdict

def word(pi):
    w = [0]*(len(pi)-1)
    for i in pi.descents():
        w[i] = 1
    return Words([0,1])(w)

@cached_function
def preimages(n):
    D = defaultdict(int)
    for pi in Permutations(n):
        D[word(pi)] += 1
    return D

def statistic(word):
    return preimages(len(word)+Integer(1))[word]
Created
Jun 08, 2016 at 13:19 by Christian Stump
Updated
Jun 08, 2016 at 13:19 by Christian Stump