185 citations · 371 across the 28 of their papers we have counts for
7 papers · 1 filter
On the inversion of in terms of associated Stirling numbers
David J. Jeffrey, Robert M. Corless, David E. G. Hare +1
The function , the solution of for and large enough, has a series expansion in terms of and , with coefficients given in terms of Sti…
Polynomials involving the floor function
Inger Johanne Håland, Donald E. Knuth
Some identities are presented that generalize the formula x^3 = 3x floor(x floor(x)) - 3 floor(x) floor(x floor(x)) + floor(x)^3 + 3 frac(x) frac(x floor(x)) + frac(x)^3 to a repre…
Two-way rounding
Donald E. Knuth
Given real numbers and a permutation~ of , we can always find $\xbar_1,...,\xbar_n\in\{0,1\}$ so that the partial sums $\xbar_1+... +\xbar…
An exact analysis of stable allocation
Donald E. Knuth
Shapley and Scarf introduced a notion of stable allocation between traders and indivisible goods, when each trader has rank-ordered each of the goods. The purpose of this note is t…
Overlapping Pfaffians
Donald E. Knuth
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 i…
Partitioned tensor products and their spectra
Donald E. Knuth
A pleasant family of graphs defined by Godsil and McKay is shown to have easily computed eigenvalues in many cases.
Aztec diamonds, checkerboard graphs, and spanning trees
Donald E. Knuth
This note derives the characteristic polynomial of a graph that represents nonjump moves in a generalized game of checkers. The number of spanning trees is also determined.