Finding total unique string combination $s(r, n) = sum_{k=1}^{r} 4^k {{n+k-2}choose{k}}$ for $1 leq r leq 4$...
$begingroup$
I have a problem in counting the total of unique string combination. Provided a string of length 5 which is composed of 4 characters (ABCD), I would like to generate all the possible string combinations from one to four characters addition excluding both end addition.
Provided that the input string is ABCDA, I want to insert additional characters: w; w and x; w, x and y; w, x, y and z into any possible combination into the input string while avoiding insertion at both ends. So that the resulting combination would be: (AwBCDA, ABwCDA, ABCwDA,..., ABCDwA); (AwxBCDA, AwBxCDA, AwBCxDA,..., ABCDwxA); (AwxyBCDA, AwxByCDA, AwxBCyDA,..., ABCDwxyA); (AwxyzBCDA, AwxyBzCDA, AwxyBCzDA,..., ABCDwxyzA). And then for each substituents w, x, y, and z are substituted into A, B, C, or D.
I solved the combination above with the formula:
$$
s(r, n) = sum_{k=1}^{r} 4^{k} {{n+k-2}choose{k}}
$$
for $1leq rleq4$. The result from the above formula contains duplicates and I would like to count all the unique strings in $s(r, n)$. I tried to solve it using the following double summation:
$$
s_{U}(r, n) = sum_{k=1}^{r}1+sum_{l=1}^{k}3^{l}{{n+k-2}choose {l}}
$$
which is rearranged to:
$$
s_{U}(r, n) = r+sum_{k=1}^{r}sum_{l=1}^{k}3^{l}{{n+k-2}choose {l}}
$$
for $1 leq r leq 4$. Please kindly check whether is the formula and also the writing is correct or not.
Thank you.
combinatorics proof-verification
edited Jan 28 at 20:34
Alex Ravsky
43.3k32583
asked Jan 9 at 7:00
Vic BrownVic Brown
213
$endgroup$
add a comment |
$begingroup$
I have a problem in counting the total of unique string combination. Provided a string of length 5 which is composed of 4 characters (ABCD), I would like to generate all the possible string combinations from one to four characters addition excluding both end addition.
Provided that the input string is ABCDA, I want to insert additional characters: w; w and x; w, x and y; w, x, y and z into any possible combination into the input string while avoiding insertion at both ends. So that the resulting combination would be: (AwBCDA, ABwCDA, ABCwDA,..., ABCDwA); (AwxBCDA, AwBxCDA, AwBCxDA,..., ABCDwxA); (AwxyBCDA, AwxByCDA, AwxBCyDA,..., ABCDwxyA); (AwxyzBCDA, AwxyBzCDA, AwxyBCzDA,..., ABCDwxyzA). And then for each substituents w, x, y, and z are substituted into A, B, C, or D.
I solved the combination above with the formula:
$$
s(r, n) = sum_{k=1}^{r} 4^{k} {{n+k-2}choose{k}}
$$
for $1leq rleq4$. The result from the above formula contains duplicates and I would like to count all the unique strings in $s(r, n)$. I tried to solve it using the following double summation:
$$
s_{U}(r, n) = sum_{k=1}^{r}1+sum_{l=1}^{k}3^{l}{{n+k-2}choose {l}}
$$
which is rearranged to:
$$
s_{U}(r, n) = r+sum_{k=1}^{r}sum_{l=1}^{k}3^{l}{{n+k-2}choose {l}}
$$
for $1 leq r leq 4$. Please kindly check whether is the formula and also the writing is correct or not.
Thank you.
combinatorics proof-verification
edited Jan 28 at 20:34
Alex Ravsky
43.3k32583
asked Jan 9 at 7:00
Vic BrownVic Brown
213
$endgroup$
add a comment |
$begingroup$
I have a problem in counting the total of unique string combination. Provided a string of length 5 which is composed of 4 characters (ABCD), I would like to generate all the possible string combinations from one to four characters addition excluding both end addition.
Provided that the input string is ABCDA, I want to insert additional characters: w; w and x; w, x and y; w, x, y and z into any possible combination into the input string while avoiding insertion at both ends. So that the resulting combination would be: (AwBCDA, ABwCDA, ABCwDA,..., ABCDwA); (AwxBCDA, AwBxCDA, AwBCxDA,..., ABCDwxA); (AwxyBCDA, AwxByCDA, AwxBCyDA,..., ABCDwxyA); (AwxyzBCDA, AwxyBzCDA, AwxyBCzDA,..., ABCDwxyzA). And then for each substituents w, x, y, and z are substituted into A, B, C, or D.
I solved the combination above with the formula:
$$
s(r, n) = sum_{k=1}^{r} 4^{k} {{n+k-2}choose{k}}
$$
for $1leq rleq4$. The result from the above formula contains duplicates and I would like to count all the unique strings in $s(r, n)$. I tried to solve it using the following double summation:
$$
s_{U}(r, n) = sum_{k=1}^{r}1+sum_{l=1}^{k}3^{l}{{n+k-2}choose {l}}
$$
which is rearranged to:
$$
s_{U}(r, n) = r+sum_{k=1}^{r}sum_{l=1}^{k}3^{l}{{n+k-2}choose {l}}
$$
for $1 leq r leq 4$. Please kindly check whether is the formula and also the writing is correct or not.
Thank you.
combinatorics proof-verification
edited Jan 28 at 20:34
Alex Ravsky
43.3k32583
asked Jan 9 at 7:00
Vic BrownVic Brown
213
$endgroup$
I have a problem in counting the total of unique string combination. Provided a string of length 5 which is composed of 4 characters (ABCD), I would like to generate all the possible string combinations from one to four characters addition excluding both end addition.
Provided that the input string is ABCDA, I want to insert additional characters: w; w and x; w, x and y; w, x, y and z into any possible combination into the input string while avoiding insertion at both ends. So that the resulting combination would be: (AwBCDA, ABwCDA, ABCwDA,..., ABCDwA); (AwxBCDA, AwBxCDA, AwBCxDA,..., ABCDwxA); (AwxyBCDA, AwxByCDA, AwxBCyDA,..., ABCDwxyA); (AwxyzBCDA, AwxyBzCDA, AwxyBCzDA,..., ABCDwxyzA). And then for each substituents w, x, y, and z are substituted into A, B, C, or D.
I solved the combination above with the formula:
$$
s(r, n) = sum_{k=1}^{r} 4^{k} {{n+k-2}choose{k}}
$$
for $1leq rleq4$. The result from the above formula contains duplicates and I would like to count all the unique strings in $s(r, n)$. I tried to solve it using the following double summation:
$$
s_{U}(r, n) = sum_{k=1}^{r}1+sum_{l=1}^{k}3^{l}{{n+k-2}choose {l}}
$$
which is rearranged to:
$$
s_{U}(r, n) = r+sum_{k=1}^{r}sum_{l=1}^{k}3^{l}{{n+k-2}choose {l}}
$$
for $1 leq r leq 4$. Please kindly check whether is the formula and also the writing is correct or not.
Thank you.
combinatorics proof-verification
combinatorics proof-verification
edited Jan 28 at 20:34
Alex Ravsky
43.3k32583
asked Jan 9 at 7:00
Vic BrownVic Brown
213
edited Jan 28 at 20:34
Alex Ravsky
43.3k32583
asked Jan 9 at 7:00
Vic BrownVic Brown
213
edited Jan 28 at 20:34
Alex Ravsky
43.3k32583
edited Jan 28 at 20:34
Alex Ravsky
43.3k32583
edited Jan 28 at 20:34
Alex Ravsky
43.3k32583
43.3k32583
asked Jan 9 at 7:00
Vic BrownVic Brown
213
asked Jan 9 at 7:00
Vic BrownVic Brown
213
asked Jan 9 at 7:00
Vic BrownVic Brown
213
213
add a comment |
add a comment |
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