gUpdZddlZddlmZddlmZddlmZddlm Z GddeZ d Z d Z d Z 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.eZdZdZedZdZdS) 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 c|jrm|tjur tjS|jr tjS|jr|jrtddddddd}|dkrt||SdSdS) Nzk should be a positive integer ,)r r r) is_NumberrInfinityis_zeroZero is_integer is_negative ValueErrorr)clskfirst_known_schur_numberss \/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/combinatorics/schur_number.pyevalzSchurNumber.eval's ; =AJz!y v < C1= C !ABBB,-!rc(J(J %Avv8;<<< = =vc|jd}|dkrtdS|dkrtdS|jr0d||dz zdz Sd|zdz dz S) Nriirr r)argsr is_Integerfunc lower_bound)selff_s rr%zSchurNumber.lower_bound4s Yq\ 773<<  774== = 9TYYrAv&&22444q8 82 1}rN)__name__ __module__ __qualname____doc__ classmethodrr%rrrr sH4 = =[ =     rrc|tjurtd|dkrtd|dkrd}n-tjtjd|zdzd}t |S)NzInput must be finiterz&n must be a non-zero positive integer.rr r)rrrmathceillogr)nmin_ks r_schur_subsets_numberr4AssAJ/000AvvABBB a $(1Q37A..// 5>>rct|tr|jstdt |}|dkrdgg}n |dkrddgg}n|dkrgdg}nddgddgg}t ||krct ||}dtt ||dz dzdzD}|dxx|z cc<t ||kc|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 numberr rr)r rrr cg|] }d|zdz Srr r-).0rs r z#schur_partition..s WWWq1Q37WWWr) isinstancerrrr4len_generate_next_listrange)r2number_of_subsetssum_free_subsetsmissed_elementss rschur_partitionrBOs+^!U9AK97888-a00AvvC5 aF8 a%II;FQF+   "3 3 3./?CCWWE#6F2G2G!A#PQTU,V,VWWW/   "3 3 3 rcg}|D]8}fd|D}fd|D}||z}||9fdtt|dzD}|||}|S)Nc,g|]}|dzk |dzS)rr-r8numberr2s rr9z'_generate_next_list..s&???vQ&(rc8g|]}|dzdz k|dzdz Sr7r-rEs rr9z'_generate_next_list..s3GGG6VAX\Q5F5F&(Q,5F5F5Frc8g|]}d|zdzkd|zdzSr7r-)r8rr2s rr9z'_generate_next_list..s.MMMQ!a1 1q rr )appendr>r<) current_listr2new_listitemtemp_1temp_2new_item last_lists ` rr=r=sH""???????GGGGTGGGF?!!!!MMMM%L(9(9!(;"<"<MMMI OOIL r)r+r/ sympy.corersympy.core.basicrsympy.core.functionrsympy.core.numbersrrr4rBr=r-rrrUs """"""((((((&&&&&&22222(222j   AAAH     r