The relationship of the Narayana numbers to those of Catalan, collected last week, is simple and direct:
Cn = N(n, 1) + N(n, 2) + N(n, 3) + … + N(n, n)
As we have seen, with the corresponding graphs, N(4, 1) + N(4, 2) + N(4, 3) + N(4, 4) = 1 + 6 + 6 + 1 = 14, which is exactly is C4 . And the relation of Narayana’s numbers to Dyck’s words is even more direct (what’s that?).
In the figure we see the three ordered rooted trees with four edges and two leaves, which together with last week’s 3 complete the possible 6, a number equal to N(4, 2). In general, there are N(n,k) rooted trees with n edges and k leaves.
The Narayana Series
Regarding the problem of the Narayana cows, our protagonists from last week, if we start with a newborn calf at the beginning of the first year, we will only have one cow for three years. In the fourth year the calf has become an adult cow and will have a calf, and another in the fifth year and another in the sixth. At the beginning of the seventh year the first calf is already fully grown, so two calves are born (that of the original cow and that of her first daughter) and so on. With this we get the sequence:
1, 1, 1, 2, 3, 4, 6, 9, 13, 19…
It is similar but not the same as Fibonacci:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
The second grows faster (you know, female rabbits are much more productive than cows) since each term is the sum of the previous two, while in the first each term is… How would you express each term in the sequence of Narayana? based on the above? What would be the twentieth link in the sequence? How many cows would we have in total after 20 years?
The problem, as stated by Narayama Pandita, assumes that all cows live through the entire process, which means the first cow will live at least 20 years, an optimistic but plausible estimate. More pessimistically, our usual commentator, Francisco Montesinos, proposes a variant in which the cows live only 7 years (see comment 9 of the previous edition). In this case, how many cows would there be after 20 years?
In an interesting article (I’ll give the hint next week) I find this table:
Does it have anything to do with the topic? What does that mean?
Have a great 2023!
Errors or omissions aside, this episode of The Science Game (number 395 in EL PAÍS) will be released on December 30th, at the gates of 2023, so it’s worth taking a look at the number as such. At first glance, it doesn’t seem very interesting: it’s not prime, it’s not perfect, it’s not polygonal, it’s not regular, it’s not factorial, it’s not Fibonacci or Catalan… It might seem sphenic because it’s the product of three prime numbers: 2023 = 7 x 17 x 17, but 17 is repeated, so neither nor. Can my attentive readers spot a less obvious quality of this number that will accompany us for 365 days? In any case, I wish you a great year in all respects.
By the way, by omitting the opening exclamation point from “A great 2023!” it’s not clear whether I’m congratulating you on the Italian New Year or alluding to the greatness of the faculty of 2023. What order is 2023!? How many zeros does it end with? What’s your first digit?
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