<content> Combinatorial </content>

312d9d91-aeeb-470e-90ea-709092491567 4 4 5 false false 0 0 decimal

Combinatorial utilities - Applications

Row Vector with n*n^r Variations with repetitions n r CPerms q 1 n range : V 0 : M 0 : w 1 n r^range : 14 mat h 1 r range c 0 : j 1 n h 1 -^range k 1 n range m 1 n r h -^range V h c c 1 + : el q k el : for for for 21 line for M 1 w el V w col : 31 line :

Row Vector with n*n! Permutations n CPerms r 1 n range : q r : M 0 : 13 mat c 1 n! range M 1 c el q r el : h n 1 - (1 range q h el q h 1 + el < break if for j h 1 + : k n : m n : 13 mat j k < q j el q k el 12 mat q k el q j el 12 mat : j j 1 + : k k 1 - : 12 mat 21 line while q m el q h el > m m 1 - : while q h el q m 1 + el 12 mat q m 1 + el q h el 12 mat : 61 line for M 31 line :

P# CU U#:set_Union(P#) U# P# 1 el 11 mat : M# 0 : c# 1 : 13 mat j# 2 : j# P# rows ≤ j# j# 1 + : U# P# j# el 1 findrows num2str 0 ≡ U# c# c# 1 + : el P# j# el : if for U# 41 line :

A function for eliminate duplictates without changing the given order of the elements

P# x# P# set_ElementOf x# P# set_ElementOf set_SetBuilder :

Q# P# P# set_Union :

O# X# r# C A# 1 X# range : n# X# : 12 mat X# GetType number ≡ A# X# : n# A# length : 12 mat cases k# 1 r# range : V# k# 11 mat : 12 mat x# Is# O# x# strrep strlen O# strlen < (: P# U# c# 1 P# cols range V# V# P# c# el k# el sort 11 mat set_Union : C Is# V# V# P# c# el k# el 11 mat set_Union : cases for V# transpose 21 line : V# n# CPerms : π Is# V# n# r# CPerms : P Is# R Is# & V# n# CPerms U# : P Is# V# n# r# CPerms U# : C Is# R Is# & V# n# CPerms U# : C Is# Unknown option error cases P# 0 : c# 1 V# cols range : P# c# el A# V# c# el el : 13 mat M# 0 : U# P# CU : c# 1 U# rows range : 13 mat M# k# c# el U# c# el k# el : M Is# P# transpose cases 81 line :

n# r# C n# IsString num2str 1 ≡ n# P π strrep r# r# r# GetType number ≡ r# length cases C n#! n# r# - (! r#! * / if 11 line :

<content> Problemas </content>

In how many ways can 3 people sit in 8 chairs so that there are no two consecutive people?

n 8 :

r 3 :

P x n r CPerms set_ElementOf x k 1 + el x k el 1 + > k 1 r 1 - product x k 1 + el x k el > k 1 r 1 - product * (1 ≡ set_SetBuilder :

P set_Cardinality 20

n r - 1 + r C 20

P transpose 25731 mat 25831 mat 24831 mat 24631 mat 24731 mat 36831 mat 46831 mat 35831 mat 26831 mat 35731 mat 13831 mat 14631 mat 13731 mat 13531 mat 13631 mat 15831 mat 16831 mat 15731 mat 14731 mat 14831 mat 120 mat

What if the chairs are at a round table?

Q x P set_ElementOf x r el n < x 1 el 1 ≡ 1 cases set_SetBuilder :

Q set_Cardinality 16

n r - 1 + r C n r - 1 - r 2 - C - 16

Q transpose 14631 mat 13731 mat 26831 mat 35731 mat 15731 mat 14731 mat 13531 mat 13631 mat 24831 mat 24631 mat 25731 mat 25831 mat 46831 mat 35831 mat 24731 mat 36831 mat 116 mat

Find all natural solutions of a + b + c = 6. This problem is equivalent to finding how to put 6 objects in 3 boxes.

n 6 :

r 3 :

S.0 PR 0 n range r C : 00031 mat 00131 mat 00231 mat 00331 mat 00431 mat 00531 mat 00631 mat 01031 mat 01131 mat 01231 mat 01331 mat 01431 mat 01531 mat 113 mat

S.0 set_Cardinality 343

P.0 x S.0 set_ElementOf x k el k 1 r sum n ≡ set_SetBuilder :

P.0 set_Cardinality 28

n r + 1 - r 1 - C 28

P.0 transpose 30331 mat 31231 mat 32131 mat 24031 mat 21331 mat 22231 mat 23131 mat 50131 mat 51031 mat 60031 mat 42031 mat 33031 mat 40231 mat 41131 mat 04231 mat 05131 mat 06031 mat 03331 mat 00631 mat 01531 mat 02431 mat 14131 mat 15031 mat 123 mat

Find the positive solutions to the previous problem. This is equivalent to not leaving any boxes empty.

P x n r - 1 + r CPerms set_ElementOf x k el k 1 r sum n ≡ set_SetBuilder :

P set_Cardinality 10

n r - r + 1 - r 1 - C 10

P transpose 23131 mat 22231 mat 31231 mat 41131 mat 32131 mat 12331 mat 11431 mat 13231 mat 21331 mat 14131 mat 110 mat

x P.0 set_ElementOf x k el k 1 r product 0 ≠ set_SetBuilder set_Cardinality 10

Now, leave at least one box empty.

P x S.0 set_ElementOf x k el k 1 r sum n ≡ (x k el k 1 r product 0 ≡ (& set_SetBuilder :

P set_Cardinality 18

P transpose 30331 mat 33031 mat 20431 mat 24031 mat 40231 mat 51031 mat 60031 mat 42031 mat 50131 mat 02431 mat 03331 mat 00631 mat 112 mat

x P.0 set_ElementOf x k el k 1 r product 0 ≡ set_SetBuilder set_Cardinality 18

Alvaro