- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000289: Binary words ⟶ ℤ

Values

0 => [2] => 10 => 2

1 => [1,1] => 11 => 3

00 => [3] => 100 => 4

01 => [2,1] => 101 => 5

10 => [1,2] => 110 => 6

11 => [1,1,1] => 111 => 7

000 => [4] => 1000 => 8

001 => [3,1] => 1001 => 9

010 => [2,2] => 1010 => 10

011 => [2,1,1] => 1011 => 11

100 => [1,3] => 1100 => 12

101 => [1,2,1] => 1101 => 13

110 => [1,1,2] => 1110 => 14

111 => [1,1,1,1] => 1111 => 15

0000 => [5] => 10000 => 16

0001 => [4,1] => 10001 => 17

0010 => [3,2] => 10010 => 18

0011 => [3,1,1] => 10011 => 19

0100 => [2,3] => 10100 => 20

0101 => [2,2,1] => 10101 => 21

0110 => [2,1,2] => 10110 => 22

0111 => [2,1,1,1] => 10111 => 23

1000 => [1,4] => 11000 => 24

1001 => [1,3,1] => 11001 => 25

1010 => [1,2,2] => 11010 => 26

1011 => [1,2,1,1] => 11011 => 27

1100 => [1,1,3] => 11100 => 28

1101 => [1,1,2,1] => 11101 => 29

1110 => [1,1,1,2] => 11110 => 30

1111 => [1,1,1,1,1] => 11111 => 31

00000 => [6] => 100000 => 32

00001 => [5,1] => 100001 => 33

00010 => [4,2] => 100010 => 34

00011 => [4,1,1] => 100011 => 35

00100 => [3,3] => 100100 => 36

00101 => [3,2,1] => 100101 => 37

00110 => [3,1,2] => 100110 => 38

00111 => [3,1,1,1] => 100111 => 39

01000 => [2,4] => 101000 => 40

01001 => [2,3,1] => 101001 => 41

01010 => [2,2,2] => 101010 => 42

01011 => [2,2,1,1] => 101011 => 43

01100 => [2,1,3] => 101100 => 44

01101 => [2,1,2,1] => 101101 => 45

01110 => [2,1,1,2] => 101110 => 46

01111 => [2,1,1,1,1] => 101111 => 47

10000 => [1,5] => 110000 => 48

10001 => [1,4,1] => 110001 => 49

10010 => [1,3,2] => 110010 => 50

10011 => [1,3,1,1] => 110011 => 51

10100 => [1,2,3] => 110100 => 52

10101 => [1,2,2,1] => 110101 => 53

10110 => [1,2,1,2] => 110110 => 54

10111 => [1,2,1,1,1] => 110111 => 55

11000 => [1,1,4] => 111000 => 56

11001 => [1,1,3,1] => 111001 => 57

11010 => [1,1,2,2] => 111010 => 58

11011 => [1,1,2,1,1] => 111011 => 59

11100 => [1,1,1,3] => 111100 => 60

11101 => [1,1,1,2,1] => 111101 => 61

11110 => [1,1,1,1,2] => 111110 => 62

11111 => [1,1,1,1,1,1] => 111111 => 63

000000 => [7] => 1000000 => 64

000001 => [6,1] => 1000001 => 65

000010 => [5,2] => 1000010 => 66

000011 => [5,1,1] => 1000011 => 67

000100 => [4,3] => 1000100 => 68

000101 => [4,2,1] => 1000101 => 69

000110 => [4,1,2] => 1000110 => 70

000111 => [4,1,1,1] => 1000111 => 71

001000 => [3,4] => 1001000 => 72

001001 => [3,3,1] => 1001001 => 73

001010 => [3,2,2] => 1001010 => 74

001011 => [3,2,1,1] => 1001011 => 75

001100 => [3,1,3] => 1001100 => 76

001101 => [3,1,2,1] => 1001101 => 77

001110 => [3,1,1,2] => 1001110 => 78

001111 => [3,1,1,1,1] => 1001111 => 79

010000 => [2,5] => 1010000 => 80

010001 => [2,4,1] => 1010001 => 81

010010 => [2,3,2] => 1010010 => 82

010011 => [2,3,1,1] => 1010011 => 83

010100 => [2,2,3] => 1010100 => 84

010101 => [2,2,2,1] => 1010101 => 85

010110 => [2,2,1,2] => 1010110 => 86

010111 => [2,2,1,1,1] => 1010111 => 87

011000 => [2,1,4] => 1011000 => 88

011001 => [2,1,3,1] => 1011001 => 89

011010 => [2,1,2,2] => 1011010 => 90

011011 => [2,1,2,1,1] => 1011011 => 91

011100 => [2,1,1,3] => 1011100 => 92

011101 => [2,1,1,2,1] => 1011101 => 93

011110 => [2,1,1,1,2] => 1011110 => 94

011111 => [2,1,1,1,1,1] => 1011111 => 95

100000 => [1,6] => 1100000 => 96

100001 => [1,5,1] => 1100001 => 97

100010 => [1,4,2] => 1100010 => 98

100011 => [1,4,1,1] => 1100011 => 99

100100 => [1,3,3] => 1100100 => 100

100101 => [1,3,2,1] => 1100101 => 101

100110 => [1,3,1,2] => 1100110 => 102

>>> Load all 511 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q^{2} + q^{3}$

$F_{2} = q^{4} + q^{5} + q^{6} + q^{7}$

$F_{3} = q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + q^{14} + q^{15}$

$F_{4} = q^{16} + q^{17} + q^{18} + q^{19} + q^{20} + q^{21} + q^{22} + q^{23} + q^{24} + q^{25} + q^{26} + q^{27} + q^{28} + q^{29} + q^{30} + q^{31}$

$F_{5} = q^{32} + q^{33} + q^{34} + q^{35} + q^{36} + q^{37} + q^{38} + q^{39} + q^{40} + q^{41} + q^{42} + q^{43} + q^{44} + q^{45} + q^{46} + q^{47} + q^{48} + q^{49} + q^{50} + q^{51} + q^{52} + q^{53} + q^{54} + q^{55} + q^{56} + q^{57} + q^{58} + q^{59} + q^{60} + q^{61} + q^{62} + q^{63}$

$F_{6} = q^{64} + q^{65} + q^{66} + q^{67} + q^{68} + q^{69} + q^{70} + q^{71} + q^{72} + q^{73} + q^{74} + q^{75} + q^{76} + q^{77} + q^{78} + q^{79} + q^{80} + q^{81} + q^{82} + q^{83} + q^{84} + q^{85} + q^{86} + q^{87} + q^{88} + q^{89} + q^{90} + q^{91} + q^{92} + q^{93} + q^{94} + q^{95} + q^{96} + q^{97} + q^{98} + q^{99} + q^{100} + q^{101} + q^{102} + q^{103} + q^{104} + q^{105} + q^{106} + q^{107} + q^{108} + q^{109} + q^{110} + q^{111} + q^{112} + q^{113} + q^{114} + q^{115} + q^{116} + q^{117} + q^{118} + q^{119} + q^{120} + q^{121} + q^{122} + q^{123} + q^{124} + q^{125} + q^{126} + q^{127}$

$F_{7} = q^{128} + q^{129} + q^{130} + q^{131} + q^{132} + q^{133} + q^{134} + q^{135} + q^{136} + q^{137} + q^{138} + q^{139} + q^{140} + q^{141} + q^{142} + q^{143} + q^{144} + q^{145} + q^{146} + q^{147} + q^{148} + q^{149} + q^{150} + q^{151} + q^{152} + q^{153} + q^{154} + q^{155} + q^{156} + q^{157} + q^{158} + q^{159} + q^{160} + q^{161} + q^{162} + q^{163} + q^{164} + q^{165} + q^{166} + q^{167} + q^{168} + q^{169} + q^{170} + q^{171} + q^{172} + q^{173} + q^{174} + q^{175} + q^{176} + q^{177} + q^{178} + q^{179} + q^{180} + q^{181} + q^{182} + q^{183} + q^{184} + q^{185} + q^{186} + q^{187} + q^{188} + q^{189} + q^{190} + q^{191} + q^{192} + q^{193} + q^{194} + q^{195} + q^{196} + q^{197} + q^{198} + q^{199} + q^{200} + q^{201} + q^{202} + q^{203} + q^{204} + q^{205} + q^{206} + q^{207} + q^{208} + q^{209} + q^{210} + q^{211} + q^{212} + q^{213} + q^{214} + q^{215} + q^{216} + q^{217} + q^{218} + q^{219} + q^{220} + q^{221} + q^{222} + q^{223} + q^{224} + q^{225} + q^{226} + q^{227} + q^{228} + q^{229} + q^{230} + q^{231} + q^{232} + q^{233} + q^{234} + q^{235} + q^{236} + q^{237} + q^{238} + q^{239} + q^{240} + q^{241} + q^{242} + q^{243} + q^{244} + q^{245} + q^{246} + q^{247} + q^{248} + q^{249} + q^{250} + q^{251} + q^{252} + q^{253} + q^{254} + q^{255}$

$F_{8} = q^{256} + q^{257} + q^{258} + q^{259} + q^{260} + q^{261} + q^{262} + q^{263} + q^{264} + q^{265} + q^{266} + q^{267} + q^{268} + q^{269} + q^{270} + q^{271} + q^{272} + q^{273} + q^{274} + q^{275} + q^{276} + q^{277} + q^{278} + q^{279} + q^{280} + q^{281} + q^{282} + q^{283} + q^{284} + q^{285} + q^{286} + q^{287} + q^{288} + q^{289} + q^{290} + q^{291} + q^{292} + q^{293} + q^{294} + q^{295} + q^{296} + q^{297} + q^{298} + q^{299} + q^{300} + q^{301} + q^{302} + q^{303} + q^{304} + q^{305} + q^{306} + q^{307} + q^{308} + q^{309} + q^{310} + q^{311} + q^{312} + q^{313} + q^{314} + q^{315} + q^{316} + q^{317} + q^{318} + q^{319} + q^{320} + q^{321} + q^{322} + q^{323} + q^{324} + q^{325} + q^{326} + q^{327} + q^{328} + q^{329} + q^{330} + q^{331} + q^{332} + q^{333} + q^{334} + q^{335} + q^{336} + q^{337} + q^{338} + q^{339} + q^{340} + q^{341} + q^{342} + q^{343} + q^{344} + q^{345} + q^{346} + q^{347} + q^{348} + q^{349} + q^{350} + q^{351} + q^{352} + q^{353} + q^{354} + q^{355} + q^{356} + q^{357} + q^{358} + q^{359} + q^{360} + q^{361} + q^{362} + q^{363} + q^{364} + q^{365} + q^{366} + q^{367} + q^{368} + q^{369} + q^{370} + q^{371} + q^{372} + q^{373} + q^{374} + q^{375} + q^{376} + q^{377} + q^{378} + q^{379} + q^{380} + q^{381} + q^{382} + q^{383} + q^{384} + q^{385} + q^{386} + q^{387} + q^{388} + q^{389} + q^{390} + q^{391} + q^{392} + q^{393} + q^{394} + q^{395} + q^{396} + q^{397} + q^{398} + q^{399} + q^{400} + q^{401} + q^{402} + q^{403} + q^{404} + q^{405} + q^{406} + q^{407} + q^{408} + q^{409} + q^{410} + q^{411} + q^{412} + q^{413} + q^{414} + q^{415} + q^{416} + q^{417} + q^{418} + q^{419} + q^{420} + q^{421} + q^{422} + q^{423} + q^{424} + q^{425} + q^{426} + q^{427} + q^{428} + q^{429} + q^{430} + q^{431} + q^{432} + q^{433} + q^{434} + q^{435} + q^{436} + q^{437} + q^{438} + q^{439} + q^{440} + q^{441} + q^{442} + q^{443} + q^{444} + q^{445} + q^{446} + q^{447} + q^{448} + q^{449} + q^{450} + q^{451} + q^{452} + q^{453} + q^{454} + q^{455} + q^{456} + q^{457} + q^{458} + q^{459} + q^{460} + q^{461} + q^{462} + q^{463} + q^{464} + q^{465} + q^{466} + q^{467} + q^{468} + q^{469} + q^{470} + q^{471} + q^{472} + q^{473} + q^{474} + q^{475} + q^{476} + q^{477} + q^{478} + q^{479} + q^{480} + q^{481} + q^{482} + q^{483} + q^{484} + q^{485} + q^{486} + q^{487} + q^{488} + q^{489} + q^{490} + q^{491} + q^{492} + q^{493} + q^{494} + q^{495} + q^{496} + q^{497} + q^{498} + q^{499} + q^{500} + q^{501} + q^{502} + q^{503} + q^{504} + q^{505} + q^{506} + q^{507} + q^{508} + q^{509} + q^{510} + q^{511}$

Description

The decimal representation of a binary word.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.

Map

to binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.