On finding a particular class of combinatorial identities

ON FINDING A P AR TICULAR CLASS OF COMBINA TORIAL IDENTI TIES KRASS IMIR Y ANKO V IORDJEV, DIMITER STOICHKO V KO V AC HEV Abstra ct. In this paper, a class of com binatorial identities is p rov ed . A metho d is used which is based o n the follo wing ru le: counting elemen ts of a given set in tw o w a ys and making equal the obtained results. This rule is know n as “counting in tw o wa y s”. The principle of inclusion and exclusion is used for obtaining a class of (0 , 1) − matrices. A mo d ification of the m etho d of “counti ng in t w o wa ys” ([1], p.2) is to obtain a general form ula ab out the num b er of elemen ts of a set, after that to consid er s ome of the sub sets of this set, and to find elemen ts of these subsets, on the one h and, by using the general formula, and on the other hand, by usin g sp ecific prop erties of the subs ets. W e w ill demonstrate this metho d by considering and pr o ving the follo wing iden tities: (1) n − 1 X s =0 ( − 1) s  n s  ( n − s ) n = n ! (2) p X s =0 ( − 1) s  2 p s  2 p − s p  2 = (2 p )! ( p !) 2 (3) n − p X s =0 ( − 1) s  n s  n − s p  k = 0 , wh ere n > pk (4) p ( k − 1) X s =0 ( − 1) s  k p s  k p − s p  k = ( k p )! ( p !) k (5) n − 1 X s =0 ( − 1) s  n s  ( n − s ) k = X ( t 1 , t 2 , . . . , t n ) , t i ≥ 1 t 1 + t 2 + · · · + t n = k k ! t 1 ! t 2 ! . . . t n ! , if n ≤ k 2000 Mathematics Subje ct Classific ation. Primary: 05A19; S econdary: 11C20 ACM-Comput ing Classific ation System (1998) : G.2.1. Key wor ds and phr ases. “counting in tw o w a ys“, (0,1)-matrix, b o olean matrix, inclu- sion and exclusion p rinciple, set. 1 2 KRASSIMIR Y ANKO V IORDJEV, DIMITER STOICHK OV KO V ACHEV What these identi ties h a v e in common is the left-hand side that can b e written by the expression: (6) R ( n × k , p ) = n − p X s =0 ( − 1) s  n s  n − s p  k and (1) is obtained with n = k and p = 1; (2) with n = 2 p and k = 2; (3) with n > pk ; (4) w ith n = kp ; (5) with p = 1. It remains to give some of th e p ossib le com bin atorial interpretatio ns of the expression (6). Bo olean (binary , or (0,1)- matrix) is a matrix w hose en tries are equal to 0 or 1. W e will show that R ( n × k , p ) giv es the n um b er of all n × k (comp osed of n ro ws and k columns ) Bo olean matrices, su c h th at in eac h column they ha ve exactly p ones, and in eac h ro w these matrices ha v e at least 1 on e. Necessary and sufficien t condition for existence of su c h matrices is 1 ≤ p ≤ n ≤ k p, as in the sp ecial case wh en n = kp , in eac h ro w w e h av e exactly 1 one, and wh en n > k p , suc h matrices d o not exist, that is, their num b er is equ al to zero (see Prop ositio n 3). Indeed, it is easy to see that the num b er r ( n × k , p ) of all n × k Bo olean matrices, th at ha v e exactly p ones in eac h column, is equal to (7) r ( n × k , p ) =  n p  k F rom the set of all n × k Boolean matrices, that h a v e exactly p ones in eac h column, w e h av e to remo ve matrices that h a v e at least one ro w of zero es, and to determine the num b er of the remaining matrices. W e will do this by using the pr inciple of inclusion and exclusion. Recall th is kno wn principle in the follo wing form: Inclusion and exclusion principle : Let M b e a finite set and let P = { p 1 , p 2 , . . . , p m } b e the s et of prop erties that can b e p ossessed b y the elemen ts of M . Denote by N ( p i 1 , p i 2 , . . . , p i s ) the num b er of elemen ts of M that p ossess th e prop erties p i 1 , p i 2 , . . . , p i s . Then the num b er N ( ∅ ) of elemen ts of M that d o not p ossess an y of the prop erties of P is giv en by the form ula: N ( ∅ ) = | M | + m X s =1 ( − 1) s X 1 ≤ i 1 n − p. Then ob viously | M | = r ( n × k, p ), and for eac h set of p rop erties { p i 1 , p i 2 , . . . , p i s } w e ha v e N ( p i 1 , p i 2 , . . . , p i s ) = r (( n − s ) × k , p ), s = 1 , 2 , . . . , n − p . Since s of n pr op erties p 1 , p 2 , . . . , p n can b e c hosen in  n s  w a ys , then n X 1 ≤ i 1 k p , ther e exists at le ast one r ow of zer o es. Prop osition 4. If B is the set of al l Bo ole an matr ic es with n r ows and k c olumns, suc h that in e ach c olumn ther e is exactly p ones, n = k p and ther e ar e no r ows of zer o es, pr ove that |B | = ( k p )! ( p !) k . Prop osition 5. If E i s the set of al l Bo ole an matric es with n r ows and k c olumns, n ≤ k , that have in e ach c olumn exactly 1 one and ther e ar e no 4 KRASSIMIR Y ANKO V IORDJEV, DIMITER STOICHK OV KO V ACHEV r ows of zer o es, pr ove that |E | = X ( t 1 , t 2 , . . . , t n ) , t i ≥ 1 t 1 + t 2 + · · · + t n = k k ! t 1 ! t 2 ! . . . t n ! . Pro ofs of Prop osition 4 and identi t y (4): Pr o of. F rom n = k p it follo w s that in eac h ro w of a matrix of the set B there is exactly 1 one and |B | = R ( n × k , p ) = R ( k p × k , p ) . Let C , C ∈ B b e an ar- bitrary matrix and the ordered n − tup le ( c 1 t 1 , c 2 t 2 , . . . , c nt n ) consists of the nonzero elemen ts of this matrix. With these elemen ts, w e asso ciate n um b ers of their columns, resp ectiv ely , namely the ordered n − tuple ( t 1 , t 2 , . . . , t n ) . Con v ersely , with the s − th elemen t t s , we can asso ciate the n onzero elemen t of ro w s of matrix C , namely c st s . If ( d 11 , d 23 , d 31 , d 43 , d 52 , d 62 ) are the nonzero elemen ts of matrix D , D ∈ B , k = 3 , p = 2 , we asso ciate the or- dered 6-tuple (1 , 3 , 1 , 3 , 2 , 2) with them. Since Bo olean matrices are uniquely determined by their n onzero entries, then the num b er of matrices of the set B is equal to the num b er of different p erm utations of ( t 1 , t 2 , . . . , t n ), that is, equal to (1 , . . . , 1 , 2 , . . . , 2 , . . . , k , . . . , k ), wh ere eac h num b er of a column (columns are k in num b er) is rep eated exactly p times, where p is the num- b er of ones in a column. Th e num b er of these p erm utations with rep etitions ([1] - the num b er of p erm utations of length n , comp osed by s different el- emen ts, r ep eated q 1 , q 2 , . . . , q s times, resp ectiv ely , is giv en by th e expr ession n ! q 1 ! q 2 ! . . . q s ! , where n = q 1 + q 2 + . . . + q s ) is equ al to n ! ( p !) k = ( k p )! ( p !) k = |B | = R ( k p × k , p ) . The pro of is completed.  F rom Pr op osition 4, with the sp ecial case wh en p = 1 , ( k = 2), we obtain Prop osition 1 (Prop osition 2). Pro ofs of Prop osition 5 and identi t y (5): Pr o of. Ea c h Boolean matrix of th e set E con tains k ones - exactly 1 one in the column and at least 1 one in eac h row. Therefore |E | = R ( n × k , 1) . If C, C ∈ E is an arbitrary matrix, the ordered k − tuple ( c r 1 1 , c r 2 2 , . . . , c r k k ) consists of its nonzero elemen ts, then let t i b e the num b er of ones in the i − th ro w, wh ere t i ≥ 1 , i = 1 , 2 , . . . , n and t 1 + t 2 + . . . + t n = k . With ma- trix C , w e can uniqu ely asso ciate th e ord er ed k − tu p le ( c r 1 1 , c r 2 2 , . . . , c r k k ) or the ordered k − tuple ( r 1 , r 2 , . . . , r k ). The n umber of Bo olean matri- ces of E , that hav e t i ones in the i − th row, is equal to the p erm utations ( r 1 , r 2 , . . . , r k ), that is, to the num b er of p erm utations of the k − tu ple (1 , . . . , 1 , . . . , 2 , . . . , 2 , . . . , n, . . . , n ), wh er e the num b er i, i = 1 , 2 , . . . , n ON FI NDING A P AR TICULAR CLASS OF COMBINA TORIAL IDENTITIES 5 is rep eated t i times. This num b er is equal to k ! t 1 ! t 2 ! . . . t n ! , and concernin g the num b er of matrices of the s et E we get |E | = X ( t 1 , t 2 , . . . , t n ) , t i ≥ 1 t 1 + t 2 + · · · + t n = k k ! t 1 ! t 2 ! . . . t n ! , where the sums are tak en o v er all p ossible expans ions of num b er k in to n nonzero terms.  Referen ces [1] M. Eigner, Combinatoria l Theory , S pringer, 1997. Dep ar tment of computer science, South-West University “Neofit Ri lski“, F ac ul ty of Ma thema tics and Na tural S ciences, 2700 B lago evgrad, Bulgaria. E-mail addr ess : iordjev@swu. bg - Krassimi r Yankov Iordjev E-mail addr ess : dkovach@abv. bg - Dimiter Stoichko v Kovachev