433 research outputs found
The Knowlton-Graham partition problem
A set partition technique that is useful for identifying wires in cables can
be recast in the language of 0--1 matrices, thereby resolving an open problem
stated by R.~L. Graham in Volume 1 of this journal. The proof involves a
construction of 0--1 matrices having row and column sums without gaps
A note on digitized angles
We study the configurations of pixels that occur when two digitized straight
lines meet each other
Overlapping Pfaffians
A combinatorial construction proves an identity for the product of the
Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices.
Several applications of this identity are followed by a brief history of
Pfaffians
Johann Faulhaber and sums of powers
Early 17th-century mathematical publications of Johann Faulhaber contain some
remarkable theorems, such as the fact that the -fold summation of
is a polynomial in when is a positive odd
number. The present paper explores a computation-based approach by which
Faulhaber may well have discovered such results, and solves a 360-year-old
riddle that Faulhaber presented to his readers. It also shows that similar
results hold when we express the sums in terms of central factorial powers
instead of ordinary powers. Faulhaber's coefficients can moreover be
generalized to factorial powers of noninteger exponents, obtaining asymptotic
series for in powers of
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