IPM, the Institute for Research in Fundamental Sciences in Tehran, Iran, com-
memorated the 20th anniversary of its founding by a major international conference
IPM 20 - Combinatorics 2009 on May 15-21, 2009. The conference was dedicated
to Reza Khosrovshahi, one of the founders of IPM and the director of its School of
Mathematics from 1996 to 2007, on the occasion of his 70th birthday. This volume
contains a collection of some of the papers presented at the conference.
Mathematics has a long history in the Iranian plateau. Between the ninth and
the fifteenth centuries, Iranian mathematicians played a central role in the develop-
ment of mathematics in the Islamic world. Mathematicians such as Khawrizmi,
Mahani, Nayrizi, Buzjani, Quhi, Karaji, Biruni, Khayyam, Tusi, and Kashani
hailed—as their names suggest—from all corners of the Iranian world, and over
a seven hundred period transformed the mathematical heritage inherited from In-
dia, Persia, and most notably Greek Alexandria. Algebra—whose name comes
from Khawrizmi’s first book on the subject— was developed and eventually be-
came a discipline distinct from Geometry with its own problems and methods. The
concept of number was enlarged to include Euclid’s magnitudes, and, by the time
of Kashani in the fifteenth century, real numbers and their decimal expansions
were used easily and productively. With applications to astronomy and geography
in mind, trigonometry and spherical geometry were systematized and developed.
Binomial coefficients and their properties (including the so-called “Pascal’s trian-
gle”) were developed and used extensively. In this period, applied mathematics
also blossomed. Mathematicians working in collaboration with artisans, architects,
and astronomers developed many practical algorithms (in fact, the word algorithm
comes from the name of Khawrizmi) and approximation procedures.
Nevertheless, after the fifteenth century, there was a marked decline in origi-
nal mathematical activity in Iran. While the study of Euclid and the classics of
Islamic mathematics continued unabated, mathematics lost the center stage. Fast
forwarding to the most recent period, Iranian mathematics started its reawaken-
ing in the years following World War II when mathematicians centered around
University of Tehran, most notably Dr. Gholamhossein Mosahab, trained a gener-
ation of mathematics educators. In turn, in the 1960s and 70s, a dedicated group
of legendary high school level math teachers disseminated a love for mathematics
among the youth. This was the period when mathematical problem solving and
mathematical books for non-experts became popular and when a number of high
school mathematics teachers developed a national reputation. The next big step in
the development of mathematics in Iran came in 1989 with the founding of IPM
(Institute for Studies in Theoretical Physics and Mathematics since renamed the
Institute for Research in Fundamental Sciences). This was complemented by the
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