## Who named Catalan numbers?

**The question.** A year ago, on this blog, I investigated Who computed Catalan numbers. Short version: it’s Euler, but many others did a lot of interesting work soon afterwards. I even made a Catalan Numbers Page with many historical and other documents. But I always assumed that the dubious honor of naming them after Eugène Catalan belongs to Netto. However, recently I saw this site which suggested that it was E.T. Bell who named the sequence. This didn’t seem right, as Bell was both a notable combinatorialist and mathematical historian. So I decided to investigate who did the deed.

**First**, I looked at Netto’s *Lehrbuch der Combinatorik* (1901). Although my German is minuscule and based on my knowledge of English and Yiddish (very little of the latter, to be sure), it was clear that Netto simply preferred counting of Catalan’s brackets to triangulations and other equivalent combinatorial interpretations. He did single out Catalan’s work, but mentioned Rodrigues’s work as well. In general, Netto wasn’t particularly careful with the the references, but in fairness neither were were most of his contemporaries. In any event, he never specifically mentioned “Catalan Zahlen”.

**Second**, I checked the above mentioned 1938 Bell’s paper in the *Annals*. As I suspected, Bell mentioned “*Catalan’s numbers*” only in passing, and not in a way to suggest that Catalan invented them. In fact, he used the term “*Euler-Segner sequence*” and provided careful historical and more recent references.

**Next** on my list was John Riordan‘s *Math Review* MR0024411, of this 1948 Motzkin’s paper. The review starts with “The Catalan numbers…”, and indeed might have been the first time this name was introduced. However, it is naive to believe that this MR moved many people to use this expression over arguably more cumbersome “Euler-Segner sequence”. In fact, Motzkin himself is very careful to cite Euler, Cayley, Kirkman, Liouville, and others. My guess is this review was immediately forgotten, but was a harbinger of things to come.

Curiously, Riordan does this again in 1964, in a *Math Review* on an English translation of a popular mathematics book by A.M. Yglom and I.M. Yaglom (published in Russian in 1954). The book mentions the sequence in the context of counting triangulations of an *n*-gon, without calling it by any name, but Riordan recognizes them and uses the term “Catalan numbers” in the review.

**The answer.** To understand what really happened, see this Ngram chart. It clearly shows that the term “Catalan numbers” took off after 1968. What happened? Google Books immediately answers – Riordan’s *Combinatorial Identities* was published in 1968 and it used “the Catalan numbers”. The term took off and became standard within a few years.

**What gives?** It seems, people really like to read books. Intentionally or unintentionally, monographs tend to standardize the definitions, notations, and names of mathematical objects. In his notes on *Mathematical writing*, Knuth mentions that the term “NP-complete problem” became standard after it was used by Aho, Hopcroft and Ullman in their famous *Data Structures and Algorithms* textbook. Similarly, Macdonald’s *Symmetric Functions and Hall Polynomials* became a standard source of names of everything in the area, just as Stanley predicted in his prescient review.

The same thing happened to Riordan’s book. Although now may be viewed as tedious, somewhat disorganized and unnecessarily simplistic (Riordan admitted to dislike differential equations, complex analysis, etc.), back in the day there was nothing better. It was lauded as “excellent and stimulating” in *P.R. Stein’s review,* which continued to say “*Combinatorial identities* is, in fact, a book that must be *read*, from cover to cover, and several times.” We are guessing it had a tremendous influence on the field and cemented the terminology and some notation.

**In conclusion.** We don’t know why Riordan chose the term “Catalan numbers”. As Motzkin’s paper shows, he clearly knew of Euler’s pioneer work. Maybe he wanted to honor Catalan for his early important work on the sequence. Or maybe he just liked the way it sounds. But Riordan clearly made a conscious decision to popularize the term back in 1948, and eventually succeeded.

UPDATE (Feb. 8, 2014) Looks like Henry Gould agrees with me (ht. Peter Luschny). He is, of course, the author of a definitive bibliography of Catalan numbers. Also, see this curious argument against naming mathematical terms after people (ht. Reinhard Zumkeller).

UPDATE** **(Aug 25, 2014)**:** I did more historical research on the subject which is now reworked into an article History of Catalan Numbers.

UPDATE (Oct 13, 2016)**:** I came across a quote from Riordan himself (see below) published in this book review. In light of our investigation, this can be read as a tacit admission that he misnamed the sequence. Note that Riordan seemed genially contrite yet unaware of the fact that Catalan learned about the sequence from Liouville who knew about Euler and Segner’s work. So the “temporary blindness” he is alleging is perhaps misaddressed…

“Nevertheless, the pursuit of originality and generality has its perils. For one

thing, the current spate of combinatorial mappings has produced the feeling

that multiplicity abounds. Perhaps the simplest example is the continuing

appearances of the Catalan numbers [..] Incidentally, these numbers

are named after E. Catalan because of a citation in Netto’s *Kombinatorik*, in

relation to perhaps the simplest bracketing problem, proposed in 1838. An

earlier appearance, which I first learned from Henry Gould, is due to the

Euler trio, Euler-Fuss-Segner, dated 1761. There are now at least forty

mappings, hence, forty diverse settings for this sequence; worse still, no end

seems in sight. In this light, the Catalan (or Euler-Fuss-Segner) originality

may be regarded as temporary blindness.”

In spite of priority, certainly there is some virtue to calling them Catalan numbers because “Eulerian numbers” would be very ambiguous! Terms like “Catalan objects”, “rational Catalan combinatorics”, etc., become immediately comprehensible thanks to this historical (mis)attribution.

I agree. The point of this post was not to express grievances about mis-named objects, but to point out that using modern tools like Ngram one can indeed figure out exactly who named them. And even for the notoriously famous sequences like Catalan numbers, the answer is surprising (at least, it was surprising to me).