A few years ago I wrote a post arguing that it is time to end ordered authorship. However that time has not yet arrived, and it appears that it is unlikely to arrive anytime soon. In the meantime, if one is writing a paper with 10 authors, a choice for authorship ordering and equal contribution designation must be made from among the almost 2 billion possibilities (1857945600 to be exact). No wonder authorship arguments are commonplace! The purpose of this short post is to explain the number 1857945600.

At first glance the enumeration of authorship orderings seems to be straightforward, namely that in a paper with *n* authors there are *n!* ways to order the authors. However this solution fails to account for designation of authors as “equal contributors”. For example, in the four author paper Structural origin of slow diffusion in protein folding, the first two authors contributed equally, and separately from that, so did the last two (as articulated via a designation of “co-corresponding” authorship). Another such example is the paper PRDM/Blimp1 downregulates expression of germinal center genes LMO2 and HGAL. Equal contribution designations can be more complex. In the recent preprint Connect-seq to superimpose molecular on anatomical neural circuit maps the first and second authors contributed equally, as did the third and fourth (though the equal contributions of the first and second authors was distinct from that of the third and fourth). Sometimes there are also more than two authors who contributed equally. In SeqVis: Visualization of compositional heterogeneity in large alignments of nucleotides the first eight authors contributed equally. A study on “equal contribution” designation in biomedical papers found that this type of designation is becoming increasingly common and can be associated with nearly every position in the byline.

To account for “equal contribution” groupings, I make the assumption that a set of authors who contributed equally must be consecutive in the authorship ordering. This assumption is certainly reasonable in the biological sciences given that there are two gradients of “contribution” (one from the front and one from the end of the authorship list), and that contributions for those in the end gradient are fundamentally distinct from those in the front. An authorship designation for a paper with *n* authors therefore consists of two separate parts: the *n!* ways to order the authors, and then the ways of designating groups of equal contribution for consecutive authors. The latter enumeration is simple: designation of equal authorship is in one-to-one correspondence with placement of dividers in the *n-1* gaps between the authors in the authorship list. In the extreme case of placement of no dividers the corresponding designation is that all authors contributed equally. Similarly, the placement of dividers between all consecutive pairs of authors corresponds to all contributions being distinct. Thus, the total number of authorship orderings/designations is given by . These numbers also enumerate **the number of ways to lace a shoe**. Other examples of objects whose enumeration results in these numbers are given in the Online Encyclopedia of Integer Sequences entry for this sequence (A002866). The first twenty numbers are:

1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280, 1857945600, 40874803200, 980995276800, 25505877196800, 714164561510400, 21424936845312000, 685597979049984000, 23310331287699456000, 839171926357180416000, 31888533201572855808000, 1275541328062914232320000.

In the case of a paper with 60 authors, the number of ways to order authors and designate equal contribution is much larger than the number of atoms in the universe. Good luck with your next consortium project!

## 7 comments

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November 3, 2018 at 6:29 am

Steven BurgessTotally agree with the comment about moving beyond author order, you might be interested in this post by a colleague about using CRedIT taxonomy in a film-credit style model https://medium.com/@devang/science-first-scientists-later-6419cbc4ac9b

November 3, 2018 at 8:52 am

gasstationwithoutpumpsWhat is needed is a separation of “authorship” from “contribution”. Papers need credits like movies do, where everyone who contributed has their credit explicitly listed (including what function they had in creating the work), but only a few people get top billing. It makes no sense to have 200 authors on a paper—they can’t all have written it, and for many the creative contribution is tenuous in the extreme (like the credits for catering or apprentice gaffer).

November 3, 2018 at 10:06 am

allenknutsonSupposedly we have such a separation (in math, anyway), with the “Acknowledgments” section as a place to acknowledge non-author contributions.

Presumably people who are grouped together shouldn’t be considered to be ordered, no? How many equivalence classes are there? I was thinking Stirling numbers but those seem a bit different.

November 3, 2018 at 10:08 am

Lior PachterYou raise a subtle but important point. Unfortunately in biology the order of “equal contributors” matters. It shouldn’t, but does. See, e.g. http://drugmonkey.scientopia.org/2012/02/28/co-first-authorship-is-a-lie-and-a-sham-and-an-embarassment-to-our-profession/

November 3, 2018 at 12:37 pm

Rick DurrettIt is certainly sensible to have the people who did most of the work at the beginning with * to indicate equal contribution, and the big shots at the end who had the ideas and money for the project at the end. In between, who knows. The simple fact is that in biology you can’t go to alphabetical order because of the importance of first author papers. A biology colleague at Cornell was surprised that Deena Schmidt who wrote three papers with me had no first author papers. .

November 4, 2018 at 2:27 am

James MartinArguably, the order of authors within equal-contribution groupings should be ignored. Counting in that way, instead of A002866, you get https://oeis.org/A000670 , starting 1, 3, 13, 75, 541, …. The OEIS page lists many interpretations. It grows somewhat slower than A002866; apparently it’s asymptotic to (1/2)*n!*(log_2(e))^(n+1) as n goes to infinity.

November 10, 2018 at 11:48 pm

Dmitry DegraveReferring to number of atoms in the Universe is not fully correct. Simply because we have no idea about size of the Universe. Everything we know is about observable Universe, which is a different concept. Size of observable Universe is limited by particle horizon.