Stirling numbers of second kind, but no two adjacent numbers in same part. Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30UTC (7:30pm US/Eastern)Stirling numbers of the second kind with constraintsBall Colouring problem clarificationSolving inequalities regarding Stirling numbersIf $S(n, n - 3) = a binom n 4 + b binom n 5 + c binom n 6$, find $a, b, c$ (where $S(n, k)$ denotes a Stirling number of the second kind)stirling numbers of second kindStirling Number of Second kind.Stirling numbers of the second kind: How to obtain a recurrence relation from a generating function?How to use bijective proof to prove $S(n+1, m+1) = sumlimits_k=m^n S(k, m) × binomnk$Find number of occurrences of $n$ in a combinationProbability of matching exactly 1 number if 3 unique numbers are picked from a set of 4 numbers, but the order of the numbers matters
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Stirling numbers of second kind, but no two adjacent numbers in same part.
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30UTC (7:30pm US/Eastern)Stirling numbers of the second kind with constraintsBall Colouring problem clarificationSolving inequalities regarding Stirling numbersIf $S(n, n - 3) = a binom n 4 + b binom n 5 + c binom n 6$, find $a, b, c$ (where $S(n, k)$ denotes a Stirling number of the second kind)stirling numbers of second kindStirling Number of Second kind.Stirling numbers of the second kind: How to obtain a recurrence relation from a generating function?How to use bijective proof to prove $S(n+1, m+1) = sumlimits_k=m^n S(k, m) × binomnk$Find number of occurrences of $n$ in a combinationProbability of matching exactly 1 number if 3 unique numbers are picked from a set of 4 numbers, but the order of the numbers matters
$begingroup$
Update: The problem has been solved. @Phicar and I individually give two transformation from $hrightarrow S$ and $Srightarrow h$, and they are inverse of each other. Any other explanation or bijective is still welcomed!
We know that the number of ways to put $n$ distinct balls (indexed $1,2,ldots,n$) into $m$ non-empty non-distinct boxes ($mleq n$) is the Stirling number of second type $S(n,m)$
We have the formula $S(n,m)=S(n-1,m-1)+mS(n-1,m)$ as well as the initial value $S(n,n)=S(n,1)=1$
Now we add the restriction that the adjacent balls should not be put into the same box(here we define $1$ and $n$ is non-adjacent),and the number of ways is $h(n,m)$
Similarly, we have $h(n,m)=h(n-1,m-1)+(m-1)h(n-1,m)$ and $h(n,n)=1,h(n,2)=1$. The only thing change here is the coefficient of the second term.
In fact, we can easily get the result that $h(n,m)=S(n-1,m-1)$
But I cannot figure out a more intuitive explanation or a bijective to show this equivalent relationship. Here I provides some basic example
$h(4,3)=S(3,2)=3$,$2,14,1$ and $3,13,1$
$h(5,3)=S(4,2)=7$,
$135,13,14,14,15,35,13$ and
$124,12,2,24,23,234,4$
combinatorics recurrence-relations stirling-numbers
New contributor
$endgroup$
add a comment |
$begingroup$
Update: The problem has been solved. @Phicar and I individually give two transformation from $hrightarrow S$ and $Srightarrow h$, and they are inverse of each other. Any other explanation or bijective is still welcomed!
We know that the number of ways to put $n$ distinct balls (indexed $1,2,ldots,n$) into $m$ non-empty non-distinct boxes ($mleq n$) is the Stirling number of second type $S(n,m)$
We have the formula $S(n,m)=S(n-1,m-1)+mS(n-1,m)$ as well as the initial value $S(n,n)=S(n,1)=1$
Now we add the restriction that the adjacent balls should not be put into the same box(here we define $1$ and $n$ is non-adjacent),and the number of ways is $h(n,m)$
Similarly, we have $h(n,m)=h(n-1,m-1)+(m-1)h(n-1,m)$ and $h(n,n)=1,h(n,2)=1$. The only thing change here is the coefficient of the second term.
In fact, we can easily get the result that $h(n,m)=S(n-1,m-1)$
But I cannot figure out a more intuitive explanation or a bijective to show this equivalent relationship. Here I provides some basic example
$h(4,3)=S(3,2)=3$,$2,14,1$ and $3,13,1$
$h(5,3)=S(4,2)=7$,
$135,13,14,14,15,35,13$ and
$124,12,2,24,23,234,4$
combinatorics recurrence-relations stirling-numbers
New contributor
$endgroup$
$begingroup$
i have added another way. You might enjoy it as well.
$endgroup$
– Phicar
Apr 15 at 23:25
add a comment |
$begingroup$
Update: The problem has been solved. @Phicar and I individually give two transformation from $hrightarrow S$ and $Srightarrow h$, and they are inverse of each other. Any other explanation or bijective is still welcomed!
We know that the number of ways to put $n$ distinct balls (indexed $1,2,ldots,n$) into $m$ non-empty non-distinct boxes ($mleq n$) is the Stirling number of second type $S(n,m)$
We have the formula $S(n,m)=S(n-1,m-1)+mS(n-1,m)$ as well as the initial value $S(n,n)=S(n,1)=1$
Now we add the restriction that the adjacent balls should not be put into the same box(here we define $1$ and $n$ is non-adjacent),and the number of ways is $h(n,m)$
Similarly, we have $h(n,m)=h(n-1,m-1)+(m-1)h(n-1,m)$ and $h(n,n)=1,h(n,2)=1$. The only thing change here is the coefficient of the second term.
In fact, we can easily get the result that $h(n,m)=S(n-1,m-1)$
But I cannot figure out a more intuitive explanation or a bijective to show this equivalent relationship. Here I provides some basic example
$h(4,3)=S(3,2)=3$,$2,14,1$ and $3,13,1$
$h(5,3)=S(4,2)=7$,
$135,13,14,14,15,35,13$ and
$124,12,2,24,23,234,4$
combinatorics recurrence-relations stirling-numbers
New contributor
$endgroup$
Update: The problem has been solved. @Phicar and I individually give two transformation from $hrightarrow S$ and $Srightarrow h$, and they are inverse of each other. Any other explanation or bijective is still welcomed!
We know that the number of ways to put $n$ distinct balls (indexed $1,2,ldots,n$) into $m$ non-empty non-distinct boxes ($mleq n$) is the Stirling number of second type $S(n,m)$
We have the formula $S(n,m)=S(n-1,m-1)+mS(n-1,m)$ as well as the initial value $S(n,n)=S(n,1)=1$
Now we add the restriction that the adjacent balls should not be put into the same box(here we define $1$ and $n$ is non-adjacent),and the number of ways is $h(n,m)$
Similarly, we have $h(n,m)=h(n-1,m-1)+(m-1)h(n-1,m)$ and $h(n,n)=1,h(n,2)=1$. The only thing change here is the coefficient of the second term.
In fact, we can easily get the result that $h(n,m)=S(n-1,m-1)$
But I cannot figure out a more intuitive explanation or a bijective to show this equivalent relationship. Here I provides some basic example
$h(4,3)=S(3,2)=3$,$2,14,1$ and $3,13,1$
$h(5,3)=S(4,2)=7$,
$135,13,14,14,15,35,13$ and
$124,12,2,24,23,234,4$
combinatorics recurrence-relations stirling-numbers
combinatorics recurrence-relations stirling-numbers
New contributor
New contributor
edited Apr 15 at 17:01
VicaYang
New contributor
asked Apr 15 at 10:16
VicaYangVicaYang
1336
1336
New contributor
New contributor
$begingroup$
i have added another way. You might enjoy it as well.
$endgroup$
– Phicar
Apr 15 at 23:25
add a comment |
$begingroup$
i have added another way. You might enjoy it as well.
$endgroup$
– Phicar
Apr 15 at 23:25
$begingroup$
i have added another way. You might enjoy it as well.
$endgroup$
– Phicar
Apr 15 at 23:25
$begingroup$
i have added another way. You might enjoy it as well.
$endgroup$
– Phicar
Apr 15 at 23:25
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
I will denote $[n]brace k=pi vdash [n]:$ the partitions of $[n]$ into $k$ blocks and i will denote $mathbbH(n,k)=pi in [n]brace k: pi text has no adjacent elements$ so that $|mathbbH(n,k)|=h(n,k).$
Consider the following function
$$varphi :[n-1]brace k-1longrightarrow mathbbH(n,k),$$
given by $varphi (pi)=gamma$ where if $pi = B_1,cdots ,B_k$ then
$gamma$ is taking each block $B$ of $pi$ and applying the algorithm find biggest $iin B$ such that $i,i-1 in B$ take $Bsetminus i-1$ and add $i$ to a new block that contains $n.$
in other words you send the elements that contradict your assumption of being adjacent to a block that contains $n.$
Example:
$$varphi (3)=colorred15|24|3$$
$$varphi (colorred12)=colorred135|2|4$$
$$varphi (2)=14|2|colorred35$$
$$varphi(1colorred23)=13|colorred25|4$$
$$varphi(2colorred34)=1|24|colorred35$$
Show that this and yours are inverse of each other.
Edit: I see you want another way. Think the following. $$mathbbH(n,k)=[n]brace ksetminus bigcup _i=1^n-1A_i,$$
where $A_i = pi in [n]brace k:i,i+1text share block$
So using inclusion-exclusion principle, you end up with
$$h(n,k)=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$
This last thing because $|A_i|=n-1brace k$ by collapsing $i$ and $i+1$ to one element. Then $|A_icap A_j|=n-2brace k$ and so on.
Independently show that $$nbrace k=sum _i = 0^n-1binomn-1in-1-ibrace k-1,$$ by choosing the elements that go with $n$ in its block. Notice that this is a binomial transformation and so you can invert it as
$$n-1 brace k-1=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$ And so $h(n,k)=n-1brace k-1.$
$endgroup$
$begingroup$
Yes, we can use the binomial transform to get the result directly
$endgroup$
– VicaYang
Apr 16 at 1:45
add a comment |
$begingroup$
I will illustrate Phicar's bijection in more detail and explain why it is invertible.
You start with a partition of $[n-1]$ into $m-1$ non-distinct parts. Let us focus on a single part. For example, when $n=12$, one part could be
$$
1,2,3,5,6,8,9,10,11
$$
Now, break this into chains of consecutive integers.
$$
1,2,3quad 5,6quad 8,9,10,11
$$
Within each chain, we will keep the highest element, remove the second highest, keeps the third highest, remove the fourth highest, etc. The removed elements will all be put into a new part with the added element, $n$.
$$
1,colorred2,3quad colorred5,6quad colorred8,9,colorred10,11
\Downarrow\1,3quad6quad
9,11quad,quad 2,5,8,10,12$$
We do this for every part. It is easy to see the result will have no consecutive integers in the same part.
Now, why is this invertible? Given a partition of $[n]$ into $m$ distinct parts with no two adjacent elements in the same part, look at the part containing $n$. Everything in that part was moved there from a different part. But it is easy to see where it was moved from; the number $k$ must have come from the part containing $k+1$. After moving all these elements back, and deleting $n$, we get a partition of $[n-1]$ into $[m-1]$ parts.
$endgroup$
$begingroup$
Yes, I realize that Phicar’s construction and mine are mutually inverse to each other.
$endgroup$
– VicaYang
Apr 15 at 16:52
add a comment |
$begingroup$
My friend HHT gives a transformation.
I use the python code to verify that my construction and @Phicar 's construction is bijective. But I still cannot provide the proof now
In $h(n,m)$, consider the boxes with $n^textth$ ball. The box contains $a_1^textth,a_2^textthldots,n^textth$. Move all the ball $a_i^textth$ to the box containing $a_i+1^textth$ until the box contains $n^textth$ ball only. Then remove the box as well as the $n^textth$ ball.
But I still cannot prove it is a bijective yet
The example:
$135,13,14,14,15,35,13$
Move balls:
$12,4,23,134,5,1,13$
Remove $5$
$12,4,23,2,124,234,24$
# assert n <= 10 for convenience, otherwise the str will be too long
# and my brute force algorithm will be too slow
import copy
def sort(arr):
for elem in arr:
elem = sorted(elem)
arr = sorted(arr, key=lambda x:x[0])
return arr
def is_valid_S(arr):
return all(arr)
def is_valid_H(arr):
if not is_valid_S(arr):
return False
for elem in arr:
for i in range(len(elem)-1):
if elem[i] + 1 == elem[i+1]:
return False
return True
# generate(5, 3, is_valid_H) or generate(4, 2, is_valid_S)
def generate(n, m, is_valid):
res = []
for i in range(m**n):
val = i
tmp = []
for i in range(m):
tmp.append([])
for idx in range(n):
tmp[val % m].append(idx+1)
val //= m
if is_valid(tmp) and sort(tmp) not in res:
res.append(sort(tmp))
return res
def H2S(m_h_arr):
h_arr = copy.deepcopy(m_h_arr)
n = max(map(max, h_arr))
idx = 0
while n not in h_arr[idx]:
idx += 1
h_arr[idx].remove(n)
for elem in h_arr[idx]:
_idx = 0
while elem + 1 not in h_arr[_idx]:
_idx += 1
h_arr[_idx].insert(h_arr[_idx].index(elem+1),elem)
del h_arr[idx]
return h_arr
def remove_adjacent(elem):
idx = len(elem) - 2
removed = []
while idx != -1:
if elem[idx] + 1 == elem[idx + 1]:
removed.append(elem[idx])
del elem[idx]
idx -= 1
return elem, removed
def S2H(m_s_arr):
s_arr = copy.deepcopy(m_s_arr)
n = max(map(max, s_arr))
removed = []
for i in range(len(s_arr)):
e, r = remove_adjacent(s_arr[i])
s_arr[i] = e
for val in r:
removed.append(val)
removed.append(n+1)
s_arr.append(sorted(removed))
return sort(s_arr)
def is_bijective(n, m, H2S, S2H):
if n > 9:
print("please set n < 10")
return
hs = generate(n, m, is_valid_H)
ss = generate(n-1, m-1, is_valid_S)
ss_ = list(map(H2S, hs))
hs_ = list(map(S2H, ss))
return all(map(lambda x:x in hs, hs_))
and all(map(lambda x:x in hs_, hs))
and all(map(lambda x:x in ss, ss_))
and all(map(lambda x:x in ss_, ss))
is_bijective(8,4,H2S,S2H)
```
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
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active
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$begingroup$
I will denote $[n]brace k=pi vdash [n]:$ the partitions of $[n]$ into $k$ blocks and i will denote $mathbbH(n,k)=pi in [n]brace k: pi text has no adjacent elements$ so that $|mathbbH(n,k)|=h(n,k).$
Consider the following function
$$varphi :[n-1]brace k-1longrightarrow mathbbH(n,k),$$
given by $varphi (pi)=gamma$ where if $pi = B_1,cdots ,B_k$ then
$gamma$ is taking each block $B$ of $pi$ and applying the algorithm find biggest $iin B$ such that $i,i-1 in B$ take $Bsetminus i-1$ and add $i$ to a new block that contains $n.$
in other words you send the elements that contradict your assumption of being adjacent to a block that contains $n.$
Example:
$$varphi (3)=colorred15|24|3$$
$$varphi (colorred12)=colorred135|2|4$$
$$varphi (2)=14|2|colorred35$$
$$varphi(1colorred23)=13|colorred25|4$$
$$varphi(2colorred34)=1|24|colorred35$$
Show that this and yours are inverse of each other.
Edit: I see you want another way. Think the following. $$mathbbH(n,k)=[n]brace ksetminus bigcup _i=1^n-1A_i,$$
where $A_i = pi in [n]brace k:i,i+1text share block$
So using inclusion-exclusion principle, you end up with
$$h(n,k)=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$
This last thing because $|A_i|=n-1brace k$ by collapsing $i$ and $i+1$ to one element. Then $|A_icap A_j|=n-2brace k$ and so on.
Independently show that $$nbrace k=sum _i = 0^n-1binomn-1in-1-ibrace k-1,$$ by choosing the elements that go with $n$ in its block. Notice that this is a binomial transformation and so you can invert it as
$$n-1 brace k-1=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$ And so $h(n,k)=n-1brace k-1.$
$endgroup$
$begingroup$
Yes, we can use the binomial transform to get the result directly
$endgroup$
– VicaYang
Apr 16 at 1:45
add a comment |
$begingroup$
I will denote $[n]brace k=pi vdash [n]:$ the partitions of $[n]$ into $k$ blocks and i will denote $mathbbH(n,k)=pi in [n]brace k: pi text has no adjacent elements$ so that $|mathbbH(n,k)|=h(n,k).$
Consider the following function
$$varphi :[n-1]brace k-1longrightarrow mathbbH(n,k),$$
given by $varphi (pi)=gamma$ where if $pi = B_1,cdots ,B_k$ then
$gamma$ is taking each block $B$ of $pi$ and applying the algorithm find biggest $iin B$ such that $i,i-1 in B$ take $Bsetminus i-1$ and add $i$ to a new block that contains $n.$
in other words you send the elements that contradict your assumption of being adjacent to a block that contains $n.$
Example:
$$varphi (3)=colorred15|24|3$$
$$varphi (colorred12)=colorred135|2|4$$
$$varphi (2)=14|2|colorred35$$
$$varphi(1colorred23)=13|colorred25|4$$
$$varphi(2colorred34)=1|24|colorred35$$
Show that this and yours are inverse of each other.
Edit: I see you want another way. Think the following. $$mathbbH(n,k)=[n]brace ksetminus bigcup _i=1^n-1A_i,$$
where $A_i = pi in [n]brace k:i,i+1text share block$
So using inclusion-exclusion principle, you end up with
$$h(n,k)=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$
This last thing because $|A_i|=n-1brace k$ by collapsing $i$ and $i+1$ to one element. Then $|A_icap A_j|=n-2brace k$ and so on.
Independently show that $$nbrace k=sum _i = 0^n-1binomn-1in-1-ibrace k-1,$$ by choosing the elements that go with $n$ in its block. Notice that this is a binomial transformation and so you can invert it as
$$n-1 brace k-1=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$ And so $h(n,k)=n-1brace k-1.$
$endgroup$
$begingroup$
Yes, we can use the binomial transform to get the result directly
$endgroup$
– VicaYang
Apr 16 at 1:45
add a comment |
$begingroup$
I will denote $[n]brace k=pi vdash [n]:$ the partitions of $[n]$ into $k$ blocks and i will denote $mathbbH(n,k)=pi in [n]brace k: pi text has no adjacent elements$ so that $|mathbbH(n,k)|=h(n,k).$
Consider the following function
$$varphi :[n-1]brace k-1longrightarrow mathbbH(n,k),$$
given by $varphi (pi)=gamma$ where if $pi = B_1,cdots ,B_k$ then
$gamma$ is taking each block $B$ of $pi$ and applying the algorithm find biggest $iin B$ such that $i,i-1 in B$ take $Bsetminus i-1$ and add $i$ to a new block that contains $n.$
in other words you send the elements that contradict your assumption of being adjacent to a block that contains $n.$
Example:
$$varphi (3)=colorred15|24|3$$
$$varphi (colorred12)=colorred135|2|4$$
$$varphi (2)=14|2|colorred35$$
$$varphi(1colorred23)=13|colorred25|4$$
$$varphi(2colorred34)=1|24|colorred35$$
Show that this and yours are inverse of each other.
Edit: I see you want another way. Think the following. $$mathbbH(n,k)=[n]brace ksetminus bigcup _i=1^n-1A_i,$$
where $A_i = pi in [n]brace k:i,i+1text share block$
So using inclusion-exclusion principle, you end up with
$$h(n,k)=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$
This last thing because $|A_i|=n-1brace k$ by collapsing $i$ and $i+1$ to one element. Then $|A_icap A_j|=n-2brace k$ and so on.
Independently show that $$nbrace k=sum _i = 0^n-1binomn-1in-1-ibrace k-1,$$ by choosing the elements that go with $n$ in its block. Notice that this is a binomial transformation and so you can invert it as
$$n-1 brace k-1=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$ And so $h(n,k)=n-1brace k-1.$
$endgroup$
I will denote $[n]brace k=pi vdash [n]:$ the partitions of $[n]$ into $k$ blocks and i will denote $mathbbH(n,k)=pi in [n]brace k: pi text has no adjacent elements$ so that $|mathbbH(n,k)|=h(n,k).$
Consider the following function
$$varphi :[n-1]brace k-1longrightarrow mathbbH(n,k),$$
given by $varphi (pi)=gamma$ where if $pi = B_1,cdots ,B_k$ then
$gamma$ is taking each block $B$ of $pi$ and applying the algorithm find biggest $iin B$ such that $i,i-1 in B$ take $Bsetminus i-1$ and add $i$ to a new block that contains $n.$
in other words you send the elements that contradict your assumption of being adjacent to a block that contains $n.$
Example:
$$varphi (3)=colorred15|24|3$$
$$varphi (colorred12)=colorred135|2|4$$
$$varphi (2)=14|2|colorred35$$
$$varphi(1colorred23)=13|colorred25|4$$
$$varphi(2colorred34)=1|24|colorred35$$
Show that this and yours are inverse of each other.
Edit: I see you want another way. Think the following. $$mathbbH(n,k)=[n]brace ksetminus bigcup _i=1^n-1A_i,$$
where $A_i = pi in [n]brace k:i,i+1text share block$
So using inclusion-exclusion principle, you end up with
$$h(n,k)=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$
This last thing because $|A_i|=n-1brace k$ by collapsing $i$ and $i+1$ to one element. Then $|A_icap A_j|=n-2brace k$ and so on.
Independently show that $$nbrace k=sum _i = 0^n-1binomn-1in-1-ibrace k-1,$$ by choosing the elements that go with $n$ in its block. Notice that this is a binomial transformation and so you can invert it as
$$n-1 brace k-1=sum _i = 0^n-1(-1)^ibinomn-1in-ibrace k.$$ And so $h(n,k)=n-1brace k-1.$
edited Apr 15 at 23:23
answered Apr 15 at 14:18
PhicarPhicar
3,03511015
3,03511015
$begingroup$
Yes, we can use the binomial transform to get the result directly
$endgroup$
– VicaYang
Apr 16 at 1:45
add a comment |
$begingroup$
Yes, we can use the binomial transform to get the result directly
$endgroup$
– VicaYang
Apr 16 at 1:45
$begingroup$
Yes, we can use the binomial transform to get the result directly
$endgroup$
– VicaYang
Apr 16 at 1:45
$begingroup$
Yes, we can use the binomial transform to get the result directly
$endgroup$
– VicaYang
Apr 16 at 1:45
add a comment |
$begingroup$
I will illustrate Phicar's bijection in more detail and explain why it is invertible.
You start with a partition of $[n-1]$ into $m-1$ non-distinct parts. Let us focus on a single part. For example, when $n=12$, one part could be
$$
1,2,3,5,6,8,9,10,11
$$
Now, break this into chains of consecutive integers.
$$
1,2,3quad 5,6quad 8,9,10,11
$$
Within each chain, we will keep the highest element, remove the second highest, keeps the third highest, remove the fourth highest, etc. The removed elements will all be put into a new part with the added element, $n$.
$$
1,colorred2,3quad colorred5,6quad colorred8,9,colorred10,11
\Downarrow\1,3quad6quad
9,11quad,quad 2,5,8,10,12$$
We do this for every part. It is easy to see the result will have no consecutive integers in the same part.
Now, why is this invertible? Given a partition of $[n]$ into $m$ distinct parts with no two adjacent elements in the same part, look at the part containing $n$. Everything in that part was moved there from a different part. But it is easy to see where it was moved from; the number $k$ must have come from the part containing $k+1$. After moving all these elements back, and deleting $n$, we get a partition of $[n-1]$ into $[m-1]$ parts.
$endgroup$
$begingroup$
Yes, I realize that Phicar’s construction and mine are mutually inverse to each other.
$endgroup$
– VicaYang
Apr 15 at 16:52
add a comment |
$begingroup$
I will illustrate Phicar's bijection in more detail and explain why it is invertible.
You start with a partition of $[n-1]$ into $m-1$ non-distinct parts. Let us focus on a single part. For example, when $n=12$, one part could be
$$
1,2,3,5,6,8,9,10,11
$$
Now, break this into chains of consecutive integers.
$$
1,2,3quad 5,6quad 8,9,10,11
$$
Within each chain, we will keep the highest element, remove the second highest, keeps the third highest, remove the fourth highest, etc. The removed elements will all be put into a new part with the added element, $n$.
$$
1,colorred2,3quad colorred5,6quad colorred8,9,colorred10,11
\Downarrow\1,3quad6quad
9,11quad,quad 2,5,8,10,12$$
We do this for every part. It is easy to see the result will have no consecutive integers in the same part.
Now, why is this invertible? Given a partition of $[n]$ into $m$ distinct parts with no two adjacent elements in the same part, look at the part containing $n$. Everything in that part was moved there from a different part. But it is easy to see where it was moved from; the number $k$ must have come from the part containing $k+1$. After moving all these elements back, and deleting $n$, we get a partition of $[n-1]$ into $[m-1]$ parts.
$endgroup$
$begingroup$
Yes, I realize that Phicar’s construction and mine are mutually inverse to each other.
$endgroup$
– VicaYang
Apr 15 at 16:52
add a comment |
$begingroup$
I will illustrate Phicar's bijection in more detail and explain why it is invertible.
You start with a partition of $[n-1]$ into $m-1$ non-distinct parts. Let us focus on a single part. For example, when $n=12$, one part could be
$$
1,2,3,5,6,8,9,10,11
$$
Now, break this into chains of consecutive integers.
$$
1,2,3quad 5,6quad 8,9,10,11
$$
Within each chain, we will keep the highest element, remove the second highest, keeps the third highest, remove the fourth highest, etc. The removed elements will all be put into a new part with the added element, $n$.
$$
1,colorred2,3quad colorred5,6quad colorred8,9,colorred10,11
\Downarrow\1,3quad6quad
9,11quad,quad 2,5,8,10,12$$
We do this for every part. It is easy to see the result will have no consecutive integers in the same part.
Now, why is this invertible? Given a partition of $[n]$ into $m$ distinct parts with no two adjacent elements in the same part, look at the part containing $n$. Everything in that part was moved there from a different part. But it is easy to see where it was moved from; the number $k$ must have come from the part containing $k+1$. After moving all these elements back, and deleting $n$, we get a partition of $[n-1]$ into $[m-1]$ parts.
$endgroup$
I will illustrate Phicar's bijection in more detail and explain why it is invertible.
You start with a partition of $[n-1]$ into $m-1$ non-distinct parts. Let us focus on a single part. For example, when $n=12$, one part could be
$$
1,2,3,5,6,8,9,10,11
$$
Now, break this into chains of consecutive integers.
$$
1,2,3quad 5,6quad 8,9,10,11
$$
Within each chain, we will keep the highest element, remove the second highest, keeps the third highest, remove the fourth highest, etc. The removed elements will all be put into a new part with the added element, $n$.
$$
1,colorred2,3quad colorred5,6quad colorred8,9,colorred10,11
\Downarrow\1,3quad6quad
9,11quad,quad 2,5,8,10,12$$
We do this for every part. It is easy to see the result will have no consecutive integers in the same part.
Now, why is this invertible? Given a partition of $[n]$ into $m$ distinct parts with no two adjacent elements in the same part, look at the part containing $n$. Everything in that part was moved there from a different part. But it is easy to see where it was moved from; the number $k$ must have come from the part containing $k+1$. After moving all these elements back, and deleting $n$, we get a partition of $[n-1]$ into $[m-1]$ parts.
answered Apr 15 at 16:25
Mike EarnestMike Earnest
28.2k22152
28.2k22152
$begingroup$
Yes, I realize that Phicar’s construction and mine are mutually inverse to each other.
$endgroup$
– VicaYang
Apr 15 at 16:52
add a comment |
$begingroup$
Yes, I realize that Phicar’s construction and mine are mutually inverse to each other.
$endgroup$
– VicaYang
Apr 15 at 16:52
$begingroup$
Yes, I realize that Phicar’s construction and mine are mutually inverse to each other.
$endgroup$
– VicaYang
Apr 15 at 16:52
$begingroup$
Yes, I realize that Phicar’s construction and mine are mutually inverse to each other.
$endgroup$
– VicaYang
Apr 15 at 16:52
add a comment |
$begingroup$
My friend HHT gives a transformation.
I use the python code to verify that my construction and @Phicar 's construction is bijective. But I still cannot provide the proof now
In $h(n,m)$, consider the boxes with $n^textth$ ball. The box contains $a_1^textth,a_2^textthldots,n^textth$. Move all the ball $a_i^textth$ to the box containing $a_i+1^textth$ until the box contains $n^textth$ ball only. Then remove the box as well as the $n^textth$ ball.
But I still cannot prove it is a bijective yet
The example:
$135,13,14,14,15,35,13$
Move balls:
$12,4,23,134,5,1,13$
Remove $5$
$12,4,23,2,124,234,24$
# assert n <= 10 for convenience, otherwise the str will be too long
# and my brute force algorithm will be too slow
import copy
def sort(arr):
for elem in arr:
elem = sorted(elem)
arr = sorted(arr, key=lambda x:x[0])
return arr
def is_valid_S(arr):
return all(arr)
def is_valid_H(arr):
if not is_valid_S(arr):
return False
for elem in arr:
for i in range(len(elem)-1):
if elem[i] + 1 == elem[i+1]:
return False
return True
# generate(5, 3, is_valid_H) or generate(4, 2, is_valid_S)
def generate(n, m, is_valid):
res = []
for i in range(m**n):
val = i
tmp = []
for i in range(m):
tmp.append([])
for idx in range(n):
tmp[val % m].append(idx+1)
val //= m
if is_valid(tmp) and sort(tmp) not in res:
res.append(sort(tmp))
return res
def H2S(m_h_arr):
h_arr = copy.deepcopy(m_h_arr)
n = max(map(max, h_arr))
idx = 0
while n not in h_arr[idx]:
idx += 1
h_arr[idx].remove(n)
for elem in h_arr[idx]:
_idx = 0
while elem + 1 not in h_arr[_idx]:
_idx += 1
h_arr[_idx].insert(h_arr[_idx].index(elem+1),elem)
del h_arr[idx]
return h_arr
def remove_adjacent(elem):
idx = len(elem) - 2
removed = []
while idx != -1:
if elem[idx] + 1 == elem[idx + 1]:
removed.append(elem[idx])
del elem[idx]
idx -= 1
return elem, removed
def S2H(m_s_arr):
s_arr = copy.deepcopy(m_s_arr)
n = max(map(max, s_arr))
removed = []
for i in range(len(s_arr)):
e, r = remove_adjacent(s_arr[i])
s_arr[i] = e
for val in r:
removed.append(val)
removed.append(n+1)
s_arr.append(sorted(removed))
return sort(s_arr)
def is_bijective(n, m, H2S, S2H):
if n > 9:
print("please set n < 10")
return
hs = generate(n, m, is_valid_H)
ss = generate(n-1, m-1, is_valid_S)
ss_ = list(map(H2S, hs))
hs_ = list(map(S2H, ss))
return all(map(lambda x:x in hs, hs_))
and all(map(lambda x:x in hs_, hs))
and all(map(lambda x:x in ss, ss_))
and all(map(lambda x:x in ss_, ss))
is_bijective(8,4,H2S,S2H)
```
New contributor
$endgroup$
add a comment |
$begingroup$
My friend HHT gives a transformation.
I use the python code to verify that my construction and @Phicar 's construction is bijective. But I still cannot provide the proof now
In $h(n,m)$, consider the boxes with $n^textth$ ball. The box contains $a_1^textth,a_2^textthldots,n^textth$. Move all the ball $a_i^textth$ to the box containing $a_i+1^textth$ until the box contains $n^textth$ ball only. Then remove the box as well as the $n^textth$ ball.
But I still cannot prove it is a bijective yet
The example:
$135,13,14,14,15,35,13$
Move balls:
$12,4,23,134,5,1,13$
Remove $5$
$12,4,23,2,124,234,24$
# assert n <= 10 for convenience, otherwise the str will be too long
# and my brute force algorithm will be too slow
import copy
def sort(arr):
for elem in arr:
elem = sorted(elem)
arr = sorted(arr, key=lambda x:x[0])
return arr
def is_valid_S(arr):
return all(arr)
def is_valid_H(arr):
if not is_valid_S(arr):
return False
for elem in arr:
for i in range(len(elem)-1):
if elem[i] + 1 == elem[i+1]:
return False
return True
# generate(5, 3, is_valid_H) or generate(4, 2, is_valid_S)
def generate(n, m, is_valid):
res = []
for i in range(m**n):
val = i
tmp = []
for i in range(m):
tmp.append([])
for idx in range(n):
tmp[val % m].append(idx+1)
val //= m
if is_valid(tmp) and sort(tmp) not in res:
res.append(sort(tmp))
return res
def H2S(m_h_arr):
h_arr = copy.deepcopy(m_h_arr)
n = max(map(max, h_arr))
idx = 0
while n not in h_arr[idx]:
idx += 1
h_arr[idx].remove(n)
for elem in h_arr[idx]:
_idx = 0
while elem + 1 not in h_arr[_idx]:
_idx += 1
h_arr[_idx].insert(h_arr[_idx].index(elem+1),elem)
del h_arr[idx]
return h_arr
def remove_adjacent(elem):
idx = len(elem) - 2
removed = []
while idx != -1:
if elem[idx] + 1 == elem[idx + 1]:
removed.append(elem[idx])
del elem[idx]
idx -= 1
return elem, removed
def S2H(m_s_arr):
s_arr = copy.deepcopy(m_s_arr)
n = max(map(max, s_arr))
removed = []
for i in range(len(s_arr)):
e, r = remove_adjacent(s_arr[i])
s_arr[i] = e
for val in r:
removed.append(val)
removed.append(n+1)
s_arr.append(sorted(removed))
return sort(s_arr)
def is_bijective(n, m, H2S, S2H):
if n > 9:
print("please set n < 10")
return
hs = generate(n, m, is_valid_H)
ss = generate(n-1, m-1, is_valid_S)
ss_ = list(map(H2S, hs))
hs_ = list(map(S2H, ss))
return all(map(lambda x:x in hs, hs_))
and all(map(lambda x:x in hs_, hs))
and all(map(lambda x:x in ss, ss_))
and all(map(lambda x:x in ss_, ss))
is_bijective(8,4,H2S,S2H)
```
New contributor
$endgroup$
add a comment |
$begingroup$
My friend HHT gives a transformation.
I use the python code to verify that my construction and @Phicar 's construction is bijective. But I still cannot provide the proof now
In $h(n,m)$, consider the boxes with $n^textth$ ball. The box contains $a_1^textth,a_2^textthldots,n^textth$. Move all the ball $a_i^textth$ to the box containing $a_i+1^textth$ until the box contains $n^textth$ ball only. Then remove the box as well as the $n^textth$ ball.
But I still cannot prove it is a bijective yet
The example:
$135,13,14,14,15,35,13$
Move balls:
$12,4,23,134,5,1,13$
Remove $5$
$12,4,23,2,124,234,24$
# assert n <= 10 for convenience, otherwise the str will be too long
# and my brute force algorithm will be too slow
import copy
def sort(arr):
for elem in arr:
elem = sorted(elem)
arr = sorted(arr, key=lambda x:x[0])
return arr
def is_valid_S(arr):
return all(arr)
def is_valid_H(arr):
if not is_valid_S(arr):
return False
for elem in arr:
for i in range(len(elem)-1):
if elem[i] + 1 == elem[i+1]:
return False
return True
# generate(5, 3, is_valid_H) or generate(4, 2, is_valid_S)
def generate(n, m, is_valid):
res = []
for i in range(m**n):
val = i
tmp = []
for i in range(m):
tmp.append([])
for idx in range(n):
tmp[val % m].append(idx+1)
val //= m
if is_valid(tmp) and sort(tmp) not in res:
res.append(sort(tmp))
return res
def H2S(m_h_arr):
h_arr = copy.deepcopy(m_h_arr)
n = max(map(max, h_arr))
idx = 0
while n not in h_arr[idx]:
idx += 1
h_arr[idx].remove(n)
for elem in h_arr[idx]:
_idx = 0
while elem + 1 not in h_arr[_idx]:
_idx += 1
h_arr[_idx].insert(h_arr[_idx].index(elem+1),elem)
del h_arr[idx]
return h_arr
def remove_adjacent(elem):
idx = len(elem) - 2
removed = []
while idx != -1:
if elem[idx] + 1 == elem[idx + 1]:
removed.append(elem[idx])
del elem[idx]
idx -= 1
return elem, removed
def S2H(m_s_arr):
s_arr = copy.deepcopy(m_s_arr)
n = max(map(max, s_arr))
removed = []
for i in range(len(s_arr)):
e, r = remove_adjacent(s_arr[i])
s_arr[i] = e
for val in r:
removed.append(val)
removed.append(n+1)
s_arr.append(sorted(removed))
return sort(s_arr)
def is_bijective(n, m, H2S, S2H):
if n > 9:
print("please set n < 10")
return
hs = generate(n, m, is_valid_H)
ss = generate(n-1, m-1, is_valid_S)
ss_ = list(map(H2S, hs))
hs_ = list(map(S2H, ss))
return all(map(lambda x:x in hs, hs_))
and all(map(lambda x:x in hs_, hs))
and all(map(lambda x:x in ss, ss_))
and all(map(lambda x:x in ss_, ss))
is_bijective(8,4,H2S,S2H)
```
New contributor
$endgroup$
My friend HHT gives a transformation.
I use the python code to verify that my construction and @Phicar 's construction is bijective. But I still cannot provide the proof now
In $h(n,m)$, consider the boxes with $n^textth$ ball. The box contains $a_1^textth,a_2^textthldots,n^textth$. Move all the ball $a_i^textth$ to the box containing $a_i+1^textth$ until the box contains $n^textth$ ball only. Then remove the box as well as the $n^textth$ ball.
But I still cannot prove it is a bijective yet
The example:
$135,13,14,14,15,35,13$
Move balls:
$12,4,23,134,5,1,13$
Remove $5$
$12,4,23,2,124,234,24$
# assert n <= 10 for convenience, otherwise the str will be too long
# and my brute force algorithm will be too slow
import copy
def sort(arr):
for elem in arr:
elem = sorted(elem)
arr = sorted(arr, key=lambda x:x[0])
return arr
def is_valid_S(arr):
return all(arr)
def is_valid_H(arr):
if not is_valid_S(arr):
return False
for elem in arr:
for i in range(len(elem)-1):
if elem[i] + 1 == elem[i+1]:
return False
return True
# generate(5, 3, is_valid_H) or generate(4, 2, is_valid_S)
def generate(n, m, is_valid):
res = []
for i in range(m**n):
val = i
tmp = []
for i in range(m):
tmp.append([])
for idx in range(n):
tmp[val % m].append(idx+1)
val //= m
if is_valid(tmp) and sort(tmp) not in res:
res.append(sort(tmp))
return res
def H2S(m_h_arr):
h_arr = copy.deepcopy(m_h_arr)
n = max(map(max, h_arr))
idx = 0
while n not in h_arr[idx]:
idx += 1
h_arr[idx].remove(n)
for elem in h_arr[idx]:
_idx = 0
while elem + 1 not in h_arr[_idx]:
_idx += 1
h_arr[_idx].insert(h_arr[_idx].index(elem+1),elem)
del h_arr[idx]
return h_arr
def remove_adjacent(elem):
idx = len(elem) - 2
removed = []
while idx != -1:
if elem[idx] + 1 == elem[idx + 1]:
removed.append(elem[idx])
del elem[idx]
idx -= 1
return elem, removed
def S2H(m_s_arr):
s_arr = copy.deepcopy(m_s_arr)
n = max(map(max, s_arr))
removed = []
for i in range(len(s_arr)):
e, r = remove_adjacent(s_arr[i])
s_arr[i] = e
for val in r:
removed.append(val)
removed.append(n+1)
s_arr.append(sorted(removed))
return sort(s_arr)
def is_bijective(n, m, H2S, S2H):
if n > 9:
print("please set n < 10")
return
hs = generate(n, m, is_valid_H)
ss = generate(n-1, m-1, is_valid_S)
ss_ = list(map(H2S, hs))
hs_ = list(map(S2H, ss))
return all(map(lambda x:x in hs, hs_))
and all(map(lambda x:x in hs_, hs))
and all(map(lambda x:x in ss, ss_))
and all(map(lambda x:x in ss_, ss))
is_bijective(8,4,H2S,S2H)
```
New contributor
edited Apr 15 at 15:58
New contributor
answered Apr 15 at 14:22
VicaYangVicaYang
1336
1336
New contributor
New contributor
add a comment |
add a comment |
VicaYang is a new contributor. Be nice, and check out our Code of Conduct.
VicaYang is a new contributor. Be nice, and check out our Code of Conduct.
VicaYang is a new contributor. Be nice, and check out our Code of Conduct.
VicaYang is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
i have added another way. You might enjoy it as well.
$endgroup$
– Phicar
Apr 15 at 23:25