A fundamental matrix equation for finite sets

Author:
H. J. Ryser

Journal:
Proc. Amer. Math. Soc. **34** (1972), 332-336

MSC:
Primary 05B20

DOI:
https://doi.org/10.1090/S0002-9939-1972-0294151-5

MathSciNet review:
0294151

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an *n*-set and let be subsets of *S*. Let *A* of size *m* by *n* be the incidence matrix for these subsets of *S*. We now regard as independent indeterminates and define . We then form the matrix product , where denotes the transpose of the matrix *A*. The symmetric matrix *Y* has in its (*i,j*) position the sum of the indeterminates in , and consequently *Y* gives us a complete description of the intersection patterns . The specialization of this basic matrix equation has been used extensively in the study of block designs. We give some other interesting applications of the matrix equation that involve subsets with various restricted intersection patterns.

**[1]**A. W. Goodman,*Set equations*, Amer. Math. Monthly**72**(1965), 607-613. MR**31**#3337. MR**0179086 (31:3337)****[2]**M. Hall, Jr.,*A problem in partitions*, Bull. Amer. Math. Soc.**47**(1941), 804-807. MR**3**, 166. MR**0005524 (3:166c)****[3]**J. B. Kelly,*Products of zero-one matrices*, Canad. J. Math.**20**(1968), 298-329. MR**37**#92. MR**0224493 (37:92)****[4]**J. H. van Lint and H. J. Ryser,*Block designs with repeated blocks*(to appear).**[5]**H. J. Ryser,*Combinatorial configurations*, SIAM J. Appl. Math.**17**(1969), 593-602. MR**41**#1559. MR**0256904 (41:1559)****[6]**-,*Intersection properties of finite sets*(to appear).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0294151-5

Keywords:
Block design,
incidence matrix,
indeterminate,
matrix equation,
set,
set intersection,
subset,
(0, 1)-matrix

Article copyright:
© Copyright 1972
American Mathematical Society