U 9%eU@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.(https://mathworld.wolfram.com/SchurNumber.html) N)S)Basic)Function)Integerc@s$eZdZdZeddZddZdS) SchurNumbera\ This function creates a SchurNumber object which is evaluated for `k \le 5` otherwise only the lower bound information can be retrieved. Examples ======== >>> from sympy.combinatorics.schur_number import SchurNumber Since S(3) = 13, hence the output is a number >>> SchurNumber(3) 13 We do not know the Schur number for values greater than 5, hence only the object is returned >>> SchurNumber(6) SchurNumber(6) Now, the lower bound information can be retrieved using lower_bound() method >>> SchurNumber(6).lower_bound() 536 cCs^|jrZ|tjkrtjS|jr"tjS|jr.|jr6tddddddd}|dkrZt||SdS) Nzk should be a positive integer ,)rrr) is_NumberrInfinityis_zeroZZero is_integerZ is_negative ValueErrorr)clskZfirst_known_schur_numbersr_/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/combinatorics/schur_number.pyeval's  zSchurNumber.evalcCsZ|jd}|dkrtdS|dkr*tdS|jrJd||ddSd|ddS) Nriir rr )argsrZ is_Integerfunc lower_bound)selfZf_rrrr4s zSchurNumber.lower_boundN)__name__ __module__ __qualname____doc__ classmethodrrrrrrr s rcCsT|tjkrtd|dkr$tdn(|dkr2d}nttd|dd}t|S)NzInput must be finiterz&n must be a non-zero positive integer.r rr )rrrmathceillogr)nZmin_krrr_schur_subsets_numberAs  r(cCst|tr|jstdt|}|dkr2dgg}n:|dkrFddgg}n&|dkr\dddgg}nddgddgg}t||krt||}ddtt||dddD}|d|7<ql|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 rcSsg|]}d|dqSr rr.0rrrr sz#schur_partition..) isinstancerrrr(len_generate_next_listrange)r'Znumber_of_subsetsZsum_free_subsetsZmissed_elementsrrrschur_partitionOs/    $r2cstg}|D]:}fdd|D}fdd|D}||}||qfddtt|dD}|||}|S)Ncs g|]}|dkr|dqS)r rr+numberr'rrr,s z'_generate_next_list..cs(g|] }|ddkr|ddqSr)rr3r5rrr,scs(g|] }d|dkrd|dqSr)rr*r5rrr,sr)appendr1r/)Z current_listr'Znew_listitemZtemp_1Ztemp_2Znew_itemZ last_listrr5rr0s  r0)r"r$Z sympy.corerZsympy.core.basicrZsympy.core.functionrZsympy.core.numbersrrr(r2r0rrrrs    5D