o o h#@sddlmZmZddlmZddlmZddlmZde de fddZ de fd d Z de fd d Z de fd dZddZddZdS))chain combinations)gcd) factorint)as_intfactorsreturncCsT|D]#}|D]\}}d}t|D]}|||}|dkr%dSqq qdS)z Check whether `n` is a nilpotent number. Note that ``factors`` is a prime factorization of `n`. This is a low-level helper for ``is_nilpotent_number``, for internal use. FT)keysitemsrange)rpqem_ru/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/combinatorics/group_numbers.py_is_nilpotent_numbers    rcCs(t|}|dkrtd|tt|S)aj Check whether `n` is a nilpotent number. A number `n` is said to be nilpotent if and only if every finite group of order `n` is nilpotent. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_nilpotent_number >>> from sympy import randprime >>> is_nilpotent_number(21) False >>> is_nilpotent_number(randprime(1, 30)**12) True References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A056867 r$n must be a positive integer, not %i)r ValueErrorrr)nrrris_nilpotent_numbers  rcCBt|}|dkrtd|t|}tdd|Do t|S)a Check whether `n` is an abelian number. A number `n` is said to be abelian if and only if every finite group of order `n` is abelian. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_abelian_number >>> from sympy import randprime >>> is_abelian_number(4) True >>> is_abelian_number(randprime(1, 2000)**2) True >>> is_abelian_number(60) False References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A051532 rrcss|]}|dkVqdS)Nr.0rrrr Vz$is_abelian_number..rrrallvaluesrrrrrris_abelian_number8  r#cCr)a Check whether `n` is a cyclic number. A number `n` is said to be cyclic if and only if every finite group of order `n` is cyclic. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_cyclic_number >>> from sympy import randprime >>> is_cyclic_number(15) True >>> is_cyclic_number(randprime(1, 2000)**2) False >>> is_cyclic_number(4) False References ========== .. [1] Pakianathan, J., Shankar, K., Nilpotent Numbers, The American Mathematical Monthly, 107(7), 631-634. .. [2] https://oeis.org/A003277 rrcss|]}|dkVqdSr Nrrrrrrwrz#is_cyclic_number..rr"rrris_cyclic_numberYr$r&csfddD}|d}tfddttdD}|D]0}t|}d}|D]tfdd||BD}||dd9}|sNnq/||7}q#|S) a| Number of groups of order `n`. where `n` is squarefree and its prime factors are ``prime_factors``. i.e., ``n == math.prod(prime_factors)`` Explanation =========== When `n` is squarefree, the number of groups of order `n` is expressed by .. math :: \sum_{d \mid n} \prod_p \frac{p^{c(p, d)} - 1}{p - 1} where `n=de`, `p` is the prime factor of `e`, and `c(p, d)` is the number of prime factors `q` of `d` such that `q \equiv 1 \pmod{p}` [2]_. The formula is elegant, but can be improved when implemented as an algorithm. Since `n` is assumed to be squarefree, the divisor `d` of `n` can be identified with the power set of prime factors. We let `N` be the set of prime factors of `n`. `F = \{p \in N : \forall q \in N, q \not\equiv 1 \pmod{p} \}, M = N \setminus F`, we have the following. .. math :: \sum_{d \in 2^{M}} \prod_{p \in M \setminus d} \frac{p^{c(p, F \cup d)} - 1}{p - 1} Practically, many prime factors are expected to be members of `F`, thus reducing computation time. Parameters ========== prime_factors : set The set of prime factors of ``n``. where `n` is squarefree. Returns ======= int : Number of groups of order ``n`` Examples ======== >>> from sympy.combinatorics.group_numbers import _holder_formula >>> _holder_formula({2}) # n = 2 1 >>> _holder_formula({2, 3}) # n = 2*3 = 6 2 See Also ======== groups_count References ========== .. [1] Otto Holder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893). http://dx.doi.org/10.1007/BF01443651 .. [2] John H. Conway, Heiko Dietrich and E.A. O'Brien, Counting groups: gnus, moas and other exotica The Mathematical Intelligencer 30, 6-15 (2008) https://doi.org/10.1007/BF02985731 cs&h|]tfddDrqS)c3s|] }|dkVqdSr%rrrr rrrsz,_holder_formula...)r )r) prime_factorsr(r s&z"_holder_formula..rc3s|]}t|VqdS)N)r)rr)Mrrrsz"_holder_formula..r csg|] }|dkr|qS)r rr'r(rr sz#_holder_formula..)r from_iterabler lenset)r)Fspowersetpsprodcr)r,r r)r_holder_formulazs?$  r7c Cs,t|}|dkrtd|t|}t|dkrt|d\}}|dkr5gd}|t|kr5||S|dkrGgd}|t|krG||S|dkrM|S|dkrSdS|d krYd S|dkrsd d|dt|ddt|dd S|d krd|dd |ddt|dddt|dd dt|ddS|dkr|dkrdSd|dd|d d|dd|dd|dd |dd|dt|dd|dd|dt|dd d|dt|ddd t|dddt|ddt|ddStdd|Drjiddd dd!ddd d"d d#d$d%d$dd d&dd'd(d)dd(dd*d d+d,d-d,d.d d/dd)dd d(d dd0d d1}||vrf||Std2t|dkrt| \}}||kr||}}||dkrdSdSt t | S)3a Number of groups of order `n`. In [1]_, ``gnu(n)`` is given, so we follow this notation here as well. Parameters ========== n : Integer ``n`` is a positive integer Returns ======= int : ``gnu(n)`` Raises ====== ValueError Number of groups of order ``n`` is unknown or not implemented. For example, gnu(`2^{11}`) is not yet known. On the other hand, gnu(12) is known to be 5, but this has not yet been implemented in this function. Examples ======== >>> from sympy.combinatorics.group_numbers import groups_count >>> groups_count(3) # There is only one cyclic group of order 3 1 >>> # There are two groups of order 10: the cyclic group and the dihedral group >>> groups_count(10) 2 See Also ======== is_cyclic_number `n` is cyclic iff gnu(n) = 1 References ========== .. [1] John H. Conway, Heiko Dietrich and E.A. O'Brien, Counting groups: gnus, moas and other exotica The Mathematical Intelligencer 30, 6-15 (2008) https://doi.org/10.1007/BF02985731 .. [2] https://oeis.org/A000001 rrr ) r r r83i i ii!lyLZ .r) r r r8r9Cii^$imMlNCr9r<='iX i ,ii i# css|]}|dkVqdSr%rrrrrrrzgroups_count..$r:(-04268 <?D )HKLPTXZ\z9Number of groups of order n is unknown or not implemented) rrrr/listr ranyr!r r7r0)rrr rA000679A090091smallrrrr groups_counts2   , :$  >    rjN) itertoolsrrsympy.external.gmpyrsympy.ntheory.factor_rsympy.utilities.miscrdictboolrrr#r&r7rjrrrrs   !! P