I had this question in an interview but I was not able to solve it.
We have a grid of 2 rows and n columns. We have to find the number of ways to sit M men and W women given no men can sit adjacent or in front of each other.
I thought of solving it by Dynamic Programming but I'm not sure how to get the recurrence relation.
I know that if I am at (0,i), I can go to (1,i+1) but I don't know how to keep track of counts of men and women so far. Can anybody help me with the recurrence relation or states of dp?
Recurrence relation
Here is one recurrence relation I could thing of.
Let's call n_ways(N, W, M, k) the number of ways to seat:
W women
and M men
in the remaining N columns
knowing there was k men in the previous column. k can only take values 0 and 1.
n_ways(N, 0, 0, k) = 1
n_ways(N, W, M, k) = 0 if M > N or W+M > 2*N
n_ways(N, W, 0, k) = ((2*N) choose W)
n_ways(N, 0, M, 0) = 2 * (N choose M)
n_ways(N, 0, M, 1) = (N choose M)
n_ways(N, W, M, 0)
= n_ways(N-1, W, M, 0) // put nobody in first column
+ 2 * n_ways(N-1, W-1, M, 0) // put 1 woman in first column
+ n_ways(N-1, W-2, M, 0) // put 2 women in first column
+ 2 * n_ways(N-1, W-1, M-1, 1) // put 1 woman & 1 man in first column
+ 2 * n_ways(N-1, W, M-1, 1) // put 1 man in first column
n_ways(N, W, M, 1)
= n_ways(N-1, W, M, 0) // put nobody in first column
+ 2 * n_ways(N-1, W-1, M, 0) // put 1 woman in first column
+ n_ways(N-1, W-2, M, 0) // put 2 women in first column
+ n_ways(N-1, W-1, M-1, 1) // put 1 woman & 1 man in first column
+ n_ways(N-1, W, M-1, 1) // put 1 man in first column
Testing with python
Since I'm too lazy to implement dynamic programming myself, I instead opted for caching using python's functools.cache.
I implemented the recurrence relation above as a cached recursive function, and got the following results:
Number of ways to seat w women and m men in n columns:
n, w, m --> ways
0, 0, 0 --> 1
0, 0, 1 --> 0
0, 1, 0 --> 0
1, 2, 0 --> 1
1, 0, 2 --> 0
1, 1, 1 --> 2
2, 2, 2 --> 2
3, 3, 3 --> 2
4, 6, 2 --> 18
10, 15, 5 --> 2364
10, 10, 10 --> 2
Here is the python code:
from functools import cache
from math import comb
#cache
def n_ways(n, w, m, k):
if w == m == 0:
return 1
elif m > n or w + m > 2 * n:
return 0
elif m == 0:
return comb(2*n, w)
elif w == 0:
return (2-k) * comb(n, m)
else:
r_0, r_w, r_ww, r_wm, r_m = (
n_ways(n-1, w, m, 0),
n_ways(n-1, w-1, m, 0),
n_ways(n-1, w-2, m, 0) if w > 1 else 0,
n_ways(n-1, w-1, m-1, 1),
n_ways(n-1, w, m-1, 1)
)
return (r_0 + r_w + r_w + r_ww + r_wm + r_m) + (1-k) * (r_wm + r_m)
if __name__=='__main__':
print(f'Number of ways to seat w women and m men in n columns:')
print(f' n, w, m --> ways')
for w, m in [(0, 0), (0, 1), (1, 0), (2, 0), (0, 2),
(1, 1), (2, 2), (3, 3), (6, 2), (15, 5),
(10, 10)]:
n = (w + m) // 2
print(f'{n:2d}, {w:2d}, {m:2d} --> ', end='')
print(n_ways(n, w, m, 0))
I'm trying to plot b against a with the following equation
2 *r* (sin[b]^2) * cos[b] * sin[a + b] == sin[a + 2*b]*((sin[a]^2) + r*(sin[b]^2))
with r at >= 0 and let's say <1000
Solve[2 cos[b] sin[b]^2 sin[a + b] == (sin[a]^2 + sin[b]^2) sin[a + 2 b], b]
However this does give me the error
Solve::nsmet: This system cannot be solved with the methods available to Solve.
How would I be able to solve this equation?
Is this what you are trying to obtain?
Edit
sol = FullSimplify#Solve[
2 r Sin[b]^2 Cos[b] Sin[a + b] == Sin[a + 2 b] (Sin[a]^2 + r Sin[b]^2), b];
Show[Table[Plot[b /. sol[[4]], {a, 3.3, 4.7}],
{r, {0, 0.1, 0.25, 0.5, 1, 2}}],
Frame -> True, Axes -> None,
PlotRange -> {Automatic, {0, Automatic}}]
I want to show that the recursion of quicksort run on best time time on n log n.
i got this recursion formula
M(0) = 1
M(1) = 1
M(n) = min (0 <= k <= n-1) {M(K) + M(n - k - 1)} + n
show that M(n) >= 1/2 (n + 1) lg(n + 1)
what i have got so far:
By induction hyposes
M(n) <= min {M(k) + M(n - k - 1} + n
focusing on the inner expresison i got:
1/2(k + 1)lg(k + 1) + 1/2(n - k)lg(n - k)
1/2lg(k + 1)^(k + 1) + 1/2lg(n - k)^(n - k)
1/2(lg(k + 1)^(k + 1) + lg(n - k)^(n - k)
1/2(lg((k + 1)^(k + 1) . (n - k)^(n - k))
But i think im doing something wrong. i think the "k" should be gonne but i cant see how this equation would cancel out all the "k". So, probably, im doing something wrong
You indeed want to get rid of k. To do this, you want to find the lower bound on the minimum of M(k) + M(n - k - 1). In general it can be arbitrarily tricky, but in this case the standard approach works: take derivative by k.
((k+1) ln(k+1) + (n-k) ln(n-k))' =
ln(k+1) + (k+1)/(k+1) - ln(n-k) - (n-k)/(n-k) =
ln((k+1) / (n-k))
We want the derivative to be 0, so
ln((k+1) / (n-k)) = 0 <=>
(k+1) / (n-k) = 1 <=>
k + 1 = n - k <=>
k = (n-1) / 2
You can check that it's indeed a local minimum.
Therefore, the best lower bound on M(k) + M(n - k - 1) (which we can get from the inductive hypothesis) is reached for k=(n-1)/2. Now you can just substitute this value instead of k, and n will be your only remaining variable.
I'm solving and plotting the equations of motion for the double pendulum using Mathematica's NDSolve.
I've successfully plotted the Angular position using a standard plot. But when I come to use the parametric plot for the position of each mass. I get no errors but simply no plot.
eqn1 = 2 th''[t] + Sin[th[t] - ph[t]] (ph'[t])^2 + Cos[th[t] - ph[t]] (ph''[t]) + (2 g/l) Sin[th[t]]
eqn2 = ph''[t] + Sin[th[t] - ph[t]] (th'[t])^2 + Cos[th[t] - ph[t]] (th''[t]) + (g/l) Sin[th[t]]
eqnA = eqn1 /. {g -> 10, l -> 1}
eqnB = eqn2 /. {g -> 10, l -> 1}
sol = NDSolve[{eqnA == 0, eqnB == 0, th[0] == 0.859, th'[0] == 0, ph[0] == 0.437, ph'[0] == 0}, {th, ph}, {t, 0, 10}]
Plot[{th[t], ph[t]} /. sol, {t, 0, 10}]
r1 = {lSin[th[t]] + lSin[ph[t]], -lCos[th[t]] - lCos[ph[t]]} /. {l -> 1, g -> 10}
ParametricPlot[r1 /. sol, {t, 0, 10}]
Replace
r1 = {lSin[th[t]] + lSin[ph[t]], -lCos[th[t]] - lCos[ph[t]]} /. {l->1, g->10}
with
r1 = {l*Sin[th[t]] + l*Sin[ph[t]], -l*Cos[th[t]] - l*Cos[ph[t]]} /. {l->1, g->10}
and your ParametricPlot should appear.
One useful trick you might remember, when any plot doesn't appear you can try replacing the plot with Table and see what it shows. Often the table of data provides the needed hint about why the plot isn't appearing.
Please, I am trying to solve definite integral and its work but the results not simplified.
F = Integrate[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, 0, 44}]
This is the result
(1/(15 (-1 + E^44)))4 (-\[Pi]^4 +
E^44 (\[Pi]^4 +
30 (-1874048 + 42592 Log[-1 + E^44] - 2904 PolyLog[2, 1/E^44] -
132 PolyLog[3, 1/E^44] - 3 PolyLog[4, 1/E^44])) +
30 (-42592 Log[-1 + E^44] +
3 (468512 + 968 PolyLog[2, 1/E^44] + 44 PolyLog[3, 1/E^44] +
PolyLog[4, 1/E^44])))
You need to convert the result to a numerical value.
F = Integrate[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, 0, 44}];
N[F]
25.9758
Visualisation
Show[Plot[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, -4, 48}],
Plot[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, 0, 44},
Filling -> Axis, PlotRange -> All]]