Metamath Proof Explorer


Table of Contents - 5.7.14. Cyclical shifts of words

A word/string can be regarded as "necklace" by connecting the two ends of the word/string together (see Wikipedia "Necklace (combinatorics)", https://en.wikipedia.org/wiki/Necklace_(combinatorics)).

Two strings are regarded as the same necklace if one string can be rotated/circularly shifted/cyclically shifted to obtain the second string. To cope with words in the sense of necklaces, the rotation/cyclic shift is defined as the basic operation, see df-csh. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 for words consisting of identical symbols and cshwshash for words having lengths which are prime numbers.

  1. ccsh
  2. df-csh
  3. cshfn
  4. cshword
  5. cshnz
  6. 0csh0
  7. cshw0
  8. cshwmodn
  9. cshwsublen
  10. cshwn
  11. cshwcl
  12. cshwlen
  13. cshwf
  14. cshwfn
  15. cshwrn
  16. cshwidxmod
  17. cshwidxmodr
  18. cshwidx0mod
  19. cshwidx0
  20. cshwidxm1
  21. cshwidxm
  22. cshwidxn
  23. cshf1
  24. cshinj
  25. repswcshw
  26. 2cshw
  27. 2cshwid
  28. lswcshw
  29. 2cshwcom
  30. cshwleneq
  31. 3cshw
  32. cshweqdif2
  33. cshweqdifid
  34. cshweqrep
  35. cshw1
  36. cshw1repsw
  37. cshwsexa
  38. 2cshwcshw
  39. scshwfzeqfzo
  40. cshwcshid
  41. cshwcsh2id
  42. cshimadifsn
  43. cshimadifsn0