Necklace (combinatorics)

src: i.ytimg.com

In combinatorics, k -now necklace long n is the string equivalent class n -similar alphabet character k , taking all rotations as equivalent. This is a structure with circularly connected beads to different colors.

Bracelets k -ary , also referred to as rotation (or free ) necklaces , is a necklace whose strings may also be equivalent under the reflection. That is, given two strings, if each is the opposite of the other then they belong to the same equivalent class. For this reason, the necklace can also be called fixed necklace to distinguish it from a turnover necklace.

Technically, one can classify a necklace as the orbit of a cyclic group action on a string of letters, and a bracelet as the orbit of the dihedral group action. This allows the application of PÃÆ'³lya calculation theorem for enumeration of necklaces and bracelets.

Video Necklace (combinatorics)

Equality class

Number of necklaces

Ada

${\ displaystyle N_ {k} (n) = {\ frac {1} {n}} \ jumlah d \ mid n} \ varphi (d ) k ^ {\ frac {n} {d}}} Â Â$

berbeda k -perhiasan panjang n , di mana ${\ displaystyle \ varphi}  ÂÂ$ adalah fungsi total Euler.

Jumlah gelang

various bracelets k -are long n , where N _k ( n ) is the number of necklaces k -long n .

Maps Necklace (combinatorics)

Example

Example necklace

If there are n beads, all of them are different, on a necklace that joins at the end, then the number of different sequences on the necklace, after allowing for rotation, is n ! / n , for n Ã, & gt; 0. This can also be expressed as ( n Ã,-1)!. This amount is less than the general case, which does not have the requirement that every bead should be different.

An intuitive justification for this can be given. If there are different n lines of objects ("beads"), the number of combinations will be n !. If the ends are combined together, the number of combinations is divided by n , since it is possible to rotate the string of beads n to n position.

Bracelet Example

If there are n beads, all different, on the bracelet combined at the end, then the number of different sequences on the bracelet, after allowing for rotation and reflection, is n ! / 2 n , for n & gt; Ã, 2. Notice that this amount is less than the general case of B _{n/n> ( n ), which does not have the requirement that every bead should be different.}

To explain this, one can start with a count for a necklace. This number can be subdivided by 2, as it is also possible to flip the bracelet.

src: thesilentwhite.com

Aperiodic Necklace

The aperiodic necklace with length n is the equivalence size class n , that is, no two different necklace rotations of the same class.

Menurut fungsi penghitungan kalung Moreau, ada

${\ displaystyle M {k} (n) = {\ frac {1} {n}} \ jumlah d \ mid n} \ mu (d ) k ^ {\ frac {n} {d}}} Â Â$

different k -the old aperiodic warfot with long n , where ? is the MÃÆ'¶bius function.

Each aperiodic necklace contains a single Lyndon word so that Lyndon's words form a representative of the aperiodic necklace.

src: images.skinnerinc.com

See also

• Lyndon word
• Inversion (discrete math)
• Necklace problem
• Necklace shares the problem
• Permutations
• Proof of Fermat's small theorem # Evidence by counting necklaces
• Forte number, binary bracelet representation of length 12 used in atonal music.

src: skinnerinc-res.cloudinary.com

References

src: skinnerinc-res.cloudinary.com

External links

• Weisstein, Eric W. "Necklace". MathWorld .
• Info about the necklace, Lyndon's words, De Bruijn order

Source of the article : Wikipedia