BinaryWordsByWeight

The binary strings of length $n$ with exactly $k$ ones.

OEISA007318

BinaryWordsByWeight(n, k)the binary strings of length

n

with exactly

k

ones

Typed signatures

Type	Package	Applies
`(integer<0..>, integer<0..>) -> indexed_collection<binary_word>`	`combinatorics`	always

Domain: Collections

Details

Count is $\binom{n}{k}$ (see Binomial).

Maps

Into and out of BinaryWord, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.

From BinaryWord

CompositionBinaryWord → CompositioninverseThe composition of m + 1 a binary word of length m cuts out.
FinsetBinaryWord → FinsetinverseThe subset of 1..m a binary word of length m marks: the positions holding 1.

To BinaryWord

BinaryWordFinset → BinaryWordinverseA subset of 1..n as its characteristic word: letter i is 1 when i is a member.
CutWordComposition → BinaryWordinverseA composition of n as the binary word of length n - 1 marking where it is cut.

31 more in FindStat, not implemented here

Mp00093DyckPath → BinaryWordInjectionGraded mapto binary word
Mp00094Composition → BinaryWordInjectionGraded mapto binary word
Mp00095IntegerPartition → BinaryWordInjectionto binary word
Mp00096BinaryWord → BinaryWordBijectionGraded mapFoata bijection
Mp00097BinaryWord → CompositionSurjectionGraded mapdelta morphism
Mp00104BinaryWord → BinaryWordBijectionGraded mapInvolutionreverse
Mp00105BinaryWord → BinaryWordBijectionGraded mapInvolutioncomplement
Mp00109Permutation → BinaryWordSurjectionGraded mapdescent word
Mp00114Permutation → BinaryWordSurjectionGraded mapconnectivity set
Mp00130Permutation → BinaryWordSurjectionGraded mapdescent tops
Mp00131Permutation → BinaryWordSurjectionGraded mapdescent bottoms
Mp00134StandardTableau → BinaryWordSurjectionGraded mapdescent word
Mp00135BinaryWord → BinaryWordBijectionGraded maprotate front-to-back
Mp00136BinaryWord → BinaryWordBijectionGraded maprotate back-to-front
Mp00158BinaryWord → BinaryWordBijectionGraded mapInvolutionalternating inverse
Mp00178BinaryWord → CompositionInjectionGraded mapto composition
Mp00200BinaryWord → BinaryWordBijectionGraded mapInvolutiontwist
Mp00224BinaryWord → BinaryWordIdempotentGraded maprunsort
Mp00234BinaryWord → BinaryWordGraded mapvalleys-to-peaks
Mp00261BinaryWord → BinaryWordGraded mapBurrows-Wheeler
Mp00262BinaryWord → Posetsposet of factors
Mp00267SignedPermutation → BinaryWordSurjectionGraded mapsigns
Mp00268BinaryWord → BinaryWordBijectionGraded mapzeros to flag zeros
Mp00269BinaryWord → BinaryWordBijectionGraded mapflag zeros to zeros
Mp00272BinaryWord → BinaryWordBijectionGraded mapGray next
Mp00273BinaryWord → BinaryWordBijectionGray previous
Mp00278BinaryWord → BinaryWordBijectionGraded maprowmotion
Mp00280BinaryWord → BinaryWordBijectionGraded mappath rowmotion
Mp00316BinaryWord → BinaryWordBijectionGraded mapinverse Foata bijection
Mp00317IntegerPartition → BinaryWordSurjectionodd parts
Mp00336BinaryWord → PermutationGraded mapstandard permutation

Examples

1 more kept as data, checked against the oracles

Basic #

$Count(BinaryWordsByWeight(4, 2)) = 6$

live ↗

\mathrm{Count}(\mathrm{BinaryWordsByWeight}(4, 2))

Out

6

The first element in enumeration order