As can be deduced from the corresponding figure from last week, the Narayana numbers, N(n, k), can be interpreted as the different lattice paths that allow us to go from (0, 0) to (2n, 0) in 2n steps covering only the diagonals of the cells and with k peaks (peaks are understood to be the upper vertices). Therefore, N(4, 4) will be the number of routes between (0, 0) and (8, 0) in 8 steps and with 4 peaks, and it is easy to see that there is only one possible path, the one in the figure , so N(4, 4) = 1.
We already knew, from the aforementioned figure from last week, that N(4, 1) = 1, N(4, 2) = 6 and N(4, 3) = 6, and, as we just saw, N(4 , 4) = 1. Therefore, the possible paths of the form N(4, k), are, in total, 1 + 6 + 6 + 1 = 14 (remember that k cannot be greater than n, so Therefore, it can only take the values 1, 2, 3 and 4). Surely the usual commentators see here a direct relationship with the Catalan numbers (what is it?).
Equally direct is the relationship of Narayana’s numbers to Dyck’s words, which we dealt with recently (can you see that relationship between numbers and words?).
Trees, words and numbers
A few weeks ago we saw the relationship between Dyck words and binary trees.
In the figure we see three ordered rooted trees (starting from an initial node called root and branching from top to bottom) with 4 edges and 2 leaves (terminal nodes from which no other starts). Can you draw any more? How are they related to the Narayana numbers?
As with Motzkin’s and Delannoy’s, there is no simple formula that gives N(n, k) as a function of n and k; it is necessary to use a double summation, or to resort to the binomial coefficients.
the other narayana
So far we have talked about TV Narayana (1930-1987), the Canadian mathematician of Indian origin who discovered the numbers that bear his name. Name that he shares with another illustrious Indian mathematician: Narayana Pandita (1340-1400), who in his treatise Ganita Kaumudi he studied what he called “additive sequences”, in which each term is obtained by sums from the previous terms, and among which is the famous Fibonacci sequence (and although Leonardo of Pisa is a couple of centuries earlier, it is highly probable that Narayana discovered the same sequence independently).
Instead of rabbits, the Indian mathematician used (mentally) cows, more in keeping with his culture, to exemplify an additive sequence, posing what has since been known as the Narayana cow problem, which reads as follows:
If cows have a calf each year, and each calf, after the three years it takes to become a mature cow, has a calf at the beginning of each year, how many cows will there be after 20 years from a first calf? ?
I invite my shrewd readers, not only to calculate how many cows there will be after 20 years, but also to make a table with the cows of each generation and draw the pertinent conclusions.
You can follow SUBJECT in Facebook, Twitter and instagramor sign up here to receive our weekly newsletter.
#Narayanas #cows