Commutation and Rearrangements

Commutation and Rearrangements
An electronic reedition of the monograph
Probl`emes combinatoires de
commutation et r´earrangements
by P. Cartier, D. Foata
with three new appendices by
D. Foata, B. Lass
and Ch. Krattenthaler
2006

Foreword
The monograph “Probl`emes combinatoires de commutation et r´earran-
gements” was originally published as no. 85 in the Springer-Verlag Lecture
Notes in Mathematics Series, back in 1969. The algebraic and combinatorial
techniques developed there have since been used in various branches of
mathematics and also computer science. The notion of partially commutative
monoid, that was first introduced for extending the MacMahon Master
Theorem to the noncommutative case, has been used in other contexts. In
particular, it has provided an appropriate mathematical model for the study
of computer parallelism. The fundamental result deals with an inversion
formula, that has been expressed in different algebraic structures, originally
the algebra of a partially commutative monoid.
It was then appropriate, with this electronic reedition of the monograph,
to have three appendices which could illustrate how that fundamental in-
version formula was implemented in other environments, explicitly and also
implicitly.
In the first appendix (“Inversions de M¨obius”) it is shown how to go from
the M¨obius inversion formula for a partially commutative monoid to the
M¨obius formula for a locally finite partially ordered set, and conversely.
In the second appendix Bodo Lass shows that by means of a simple
specialization of the variables the fundamental inversion formula provides
a noncommutative version of the celebrated chromatic polynomial identity
for graphs : (−1)|V|χG(−1) = a(G).
The third appendix, written by Christian Krattenthaler, presents Vien-
not’s theory of heaps of pieces, a theory that has been very fruitful in the
combinatorial theory of orthogonal polynomials and in the calculation of
multivariable generating functions for polyominoes. The equivalence of the
theory of heaps and the theory of partially commutative monoids is explicitly
established.

COMMUTATION AND REARRANGEMENTS
P. Cartier, D. Foata :
Probl`emes combinatoires de commutation et r´earrangements, 54 pages.
D. Foata :
InversionsdeM¨obius,5pages ...................................p.55
B. Lass :
Le polynˆome chromatique, 2 pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 61
Ch. Krattenthaler :
The theory of heaps and the Cartier-Foata monoid, 11 pages . . . . . . . . p. 63

Lecture Notes in
Mathematics
A collection of informal reports and seminars
Edited by A. Dold, Heidelberg and B. Eckmann, Z¨urich
Series : Institut de Math´ematique, Universit´e de Strasbourg
Adviser : P. A. Meyer
85
P. Cartier •D. Foata
Universit´e de Strasbourg
Probl`emes combinatoires de
commutation et r´earrangements
Springer-Verlag
Berlin •Heidelberg •New York 1969