On the volume of the polytope of doubly stochastic matrices

Clara S. Chan; David P. Robbins

Home/Archive/arXiv

arXiv · math/9806076

On the volume of the polytope of doubly stochastic matrices

Clara S. Chan·David P. Robbins

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the calculation of the volume of the polytope B_n of n by n doubly stochastic matrices; that is, the set of real non-negative matrices with all row and column sums equal to one. We describe two methods. The first involves a decomposition of the polytope into simplices. The second involves the enumeration of ``magic squares'', i.e., n by n non-negative integer matrices whose rows and columns all sum to the same integer. We have used the first method to confirm the previously known values through n=7. This method can also be used to compute the volumes of faces of B_n. For example, we have observed that the volume of a particular face of B_n appears to be a product of Catalan numbers. We have used the second method to find the volume for n=8, which we believe was not previously known.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Clara S. Chan, David P. Robbins. 1998-06-13. On the volume of the polytope of doubly stochastic matrices. https://arxiv.org/abs/math/9806076

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Domination in Johnson graphs J(n, 3) for odd n

In 2025 Cornet, Dravec, and Torres determined the domination number $γ(J(n,3))$ of the Johnson graph for every even $n \ge 6$, expressing it as a closed form $ϕ_n$ in terms of Fort-Hedlund covering numbers, and conjectured the same value for odd $n$. We prove this conjecture: $γ(J(n,3)) = ϕ_n$ for every odd $n \ge 7$. A dominating set of $J(n,3)$ is the same as a lotto design $\mathrm{LD}(n,3,3,2)$, and the lotto numbers $L(n,3,3,2)$, and hence $γ(J(n,3))$, were already determined for every $n$ by Bate (1978) and by Brouwer (1981); a short degree argument of Füredi, Székely, and Zubor (1996) gives the same lower bound for all $n \not\equiv 9, 10 \pmod{12}$. The contribution of this paper is therefore a new proof of the odd case, carried out within the bijective framework of Cornet, Dravec, and Torres.

math.CO↗

arXiv2026

HJ numbers revisited

Hales-Jewett theorem is well known. We prove a primitive recursive bound, in fact an $\clE^5$ one, for the density version. In preparation we deal with what was the main induction step in the old proof of Hales-Jewett as a result in its own right with better bounds. Also in improving our bounds.

math.CO↗

arXiv2026

Adjacency-degree algebras and spectral determination of graphs

Let $G$ be a graph with adjacency matrix $A$ and degree matrix $D$. Applying $A$ and $D$ repeatedly to the all-ones vector produces a space $M_G$ of functions on the vertices, and these functions are constant on the orbits of the automorphism group. We call $G$ AD-observable if $M_G$ contains the indicator vector of every orbit, and AD-separable if the functions in $M_G$ separate the orbits. Forests are AD-observable, by McKay's leaf-pruning argument. Our main result is that for every cograph, $M_G$ is spanned by the indicator vectors of the cells of the stable partition of color refinement. Hence a cograph is AD-observable, or equivalently AD-separable, exactly when color refinement finds its orbits, and we describe these cographs by their cotrees. We also show that two cographs with the same number of vertices and the same homomorphism counts from all caterpillars have the same counts from all trees. The main result does not extend to distance-hereditary graphs: we give two such graphs on nine vertices that are AD-separable but not AD-observable.

math.CO↗