Combinatorics
Program solutions for generating combinations, variations and permutations for any number of variables are shown as very useful to apply in various specialistic purposes... At these pages, for example, it makes the basic framework in the program of Factor analysis and the Program for findig the nonattacking pieces on the chess board with "backtrack" method, and also well it served at the Program for homogenezing meal categorized by various parameters of the wheat in the silo cells!
Tetszőleges számú változók kombinációit, variációit valamint permutációit eredményező programmegoldások nagyon eredményesen alkalmazhatók a legkülönbözőbb speciális szükségletekre... Az itt bemutatott programoknál pl. alapvázát képezik úgy a Faktoranalízisben, mint a "Backtrack" módszert felhasználó sakkfigurákra vonatkozó programban is, de ugyanolyan jónak mutatkozott a liszt homogenizációs problémájának megoldására is a silókban levő búza különböző paramétereinek függvényében!
Programska reenja za generisanje kombinacija, varijacija ili permutacija za proizvoljan broj varijabli se pokazuju kao vrlo korisne u primeni najrazlicitijih specijalistickih potreba... Na ovim stranicama npr. one cine osnovni kostur programa Faktorska analiza kao i Programa za nalaenje svih nenapadajucih ahovskih figura po takozvanoj "backtrack" metodi, a isto je tako dobro posluio i u Programu za homogenizaciju brana po raznim parametrima penice u silo celijama!
Listing for generating COMBINATIONS with minimal required codes in BASIC:
The similar solution but in structural form in ZIM:
Captured screen from chess table analysis:
Download program Backtrack_V2
The original "Backtrack_V2" combinatorial program represents an easy way for enumerating (k) nonattacking chess pieces on nXn sized rectangular boards. Although it's originally intended for demonstration of the backtracking process, it's really uses by itself a much more sophisticated method, a generator of combinations and permutations, which gives exatly the required number of the rook positions! By filtering out the diagonal attacks we can simply get the satisfactory queen positions! This technique is also noticed and rather used by Raymond Hettingers (MIT, 2009) in his Python solutions. The bishop attacks, however, can't produce directly "in passing" by this way, but the obtained results for bishops equals to of what V.Kotesovec called as "composite pieces: bishop + semirook" (example.: A185056 OEIS)
Release notes to the enhanced Backtrack_V4 Pro version:
The Backtrack_V5 Pro version shows all previous enhanced features on a simple menu.
The Backtrack_V6 Pro version is specially worked out for new opportunities of enumerating the diagonally attacked positions by various criteria, but is not publicly presented.
The Backtrack_V7 Pro version incorporates all possibilities of the previous two versions. So now, in addition to finding all nonattacking positions, there are also enabled enumeration of such queen positions where they make various diagonal attacks with each other, and the number of these cases are shown according to columns allocated by the first Queen!
n 
Counting results of exatly 3 diagonal attacks made by 3 queens shown by columns 
Sums  
3 
2 
2  
4 
8 
4 






12 
5 
20 
14 
6 





40 
6 
40 
32 
20 
8 




100 
7 
70 
60 
44 
26 
10 



210 
8 
112 
100 
80 
56 
32 
12 


392 
9 
168 
154 
130 
100 
68 
38 
14 

672 
10 
240 
224 
196 
160 
120 
80 
44 
16 
1080 
...  ...  ...  
A007290  A159920 
2*A006503 
2*A060488 
A017617  A016933  A008911 
References:http://www.dejanristanovic.com/refer/kombin.htm
 Copyright: A.Pinter, 1994 
This webpage updated Oct 10, 2015