o 8Va @shdZddlZddlmZddlmZddlmZddlm Z GdddeZ d d Z d d Z d dZ dS)z The Schur number S(k) is the largest integer n for which the interval [1,n] can be partitioned into k sum-free sets.(http://mathworld.wolfram.com/SchurNumber.html) N)S)Basic)Function)Integerc@s$eZdZdZeddZddZdS) SchurNumberaX This function creates a SchurNumber object which is evaluated for k <= 4 otherwise only the lower bound information can be retrieved. Examples ======== >>> from sympy.combinatorics.schur_number import SchurNumber Since S(3) = 13, hence the output is a number >>> SchurNumber(3) 13 We don't know the schur number for values greater than 4, hence only the object is returned >>> SchurNumber(6) SchurNumber(6) Now, the lower bound information can be retrieved using lower_bound() method >>> SchurNumber(6).lower_bound() 364 cCs^|jr+|tjur tjS|jrdS|jr|jrtdddddd}|dkr-t||SdSdS)Nrzk should be a positive integer ,)rr) is_NumberrInfinityZis_zero is_integerZ is_negative ValueErrorr)clskZfirst_known_schur_numbersrB/usr/lib/python3/dist-packages/sympy/combinatorics/schur_number.pyeval's   zSchurNumber.evalcCs|jd}d|ddS)Nrr rr )args)selfZf_rrr lower_bound4s zSchurNumber.lower_boundN)__name__ __module__ __qualname____doc__ classmethodrrrrrrr s   rcCsX|tjur td|dkrtd|dkrd}t|Sttd|dd}t|S)NzInput must be finiterz&n must be a non-zero positive integer.r rr )rrrmathZceillogr)nZmin_krrr_schur_subsets_number9s r!cCst|tr |js tdt|}|dkrdgg}n|dkr#ddgg}n|dkr-gdg}nddgddgg}t||kr`t||}ddtt||dddD}|d |7<t||ks;|S) a This function returns the partition in the minimum number of sum-free subsets according to the lower bound given by the Schur Number. Parameters ========== n: a number n is the upper limit of the range [1, n] for which we need to find and return the minimum number of free subsets according to the lower bound of schur number Returns ======= List of lists List of the minimum number of sum-free subsets Notes ===== It is possible for some n to make the partition into less subsets since the only known Schur numbers are: S(1) = 1, S(2) = 4 , S(3) = 13, S(4) = 44. e.g for n = 44 the lower bound from the function above is 5 subsets but it has been proven that can be done with 4 subsets. Examples ======== For n = 1, 2, 3 the answer is the set itself >>> from sympy.combinatorics.schur_number import schur_partition >>> schur_partition(2) [[1, 2]] For n > 3, the answer is the minimum number of sum-free subsets: >>> schur_partition(5) [[3, 2], [5], [1, 4]] >>> schur_partition(8) [[3, 2], [6, 5, 8], [1, 4, 7]] zInput value must be a numberrr r )rr r rcSsg|]}d|dqSr rr.0rrrr sz#schur_partition..) isinstancerr rr!len_generate_next_listrange)r Znumber_of_subsetsZsum_free_subsetsZmissed_elementsrrrschur_partitionGs /     $ r+csvg}|D]}fdd|D}fdd|D}||}||qfddtdt|dD}|||}|S)Ncs g|] }|dkr|dqS)r rr$Znumberr rrr%s z'_generate_next_list..cs(g|]}|ddkr|ddqSr"rr,r-rrr%(cs(g|]}d|dkrd|dqSr"rr#r-rrr%r.rr)appendr*r()Z current_listr Znew_listitemZtemp_1Ztemp_2Znew_itemZ last_listrr-rr)s   r))rrZ sympy.corerZsympy.core.basicrZsympy.core.functionrZsympy.core.numbersrrr!r+r)rrrrs    - D