Finding total unique combination for character deletion in a string












1












$begingroup$


Is there any general formula to determine a unique combination for character deletion in a string comprised of 4 characters? Let's say that I have $str1 = {ABCDA}$ and I want to delete from one up to four position in that string. Thus all the combination for deletion up to four position of str1 would be:
$$
s(1, str1) = BCDA, ACDA, ABDA, ABCA, ABCD = 5
$$
$$
s(2, str1) = CDA, BDA, BCA, BCD, ADA, ACA, ACD, ABA, ABD, ABC = 10
$$
$$
s(3, str1) = DA, CA, CD, BA, BD, BC, AA, AD, AC, AB = 10
$$
$$
s(4, str1) = A, D, C, B, A = 5
$$
and gives 30 in total, and only 29 unique strings. In general, the total combination could be solved using the formula $sum_{k=1}^{r} {{n}choose{k}}$. However I could not find a formula to calculate total unique combination from a given string since the relative position of characters determine the total unique combination. Let's say that another 5-character long string is str2 = {AAABC}, then the number of unique sequences will be different:
$$
s(1, str2) = AABC, AABC, AABC, AAAC, AAAB
$$
$$
s(2, str2) = ABC, ABC, AAC, AAB, ABC, AAC, AAB, AAC, AAB, AAA
$$
$$
s(3, str2) = BCA, AC, AB, AC, AB, AA, AC, AB, AA, AA
$$
$$
s(4, str3) = C, B, A, A, A
$$
The total combinations is still 30, but the unique combinations reduce to 14. Is there any way to count such unique combination for any string with length n composed of four characters (A, B, C, and D)?






combinatorics







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  • $begingroup$
    Answered on stackoverflow: stackoverflow.com/questions/50702755/…, stackoverflow.com/questions/5151483/….
    $endgroup$
    – Yuval Filmus
    Jan 11 at 9:08










  • $begingroup$
    Thanks for the above link @YuvalFilmus. However, I need the mathematical formula which could be applied to a general problem of string deletion.
    $endgroup$
    – Vic Brown
    Jan 14 at 2:23
















1












$begingroup$


Is there any general formula to determine a unique combination for character deletion in a string comprised of 4 characters? Let's say that I have $str1 = {ABCDA}$ and I want to delete from one up to four position in that string. Thus all the combination for deletion up to four position of str1 would be:
$$
s(1, str1) = BCDA, ACDA, ABDA, ABCA, ABCD = 5
$$
$$
s(2, str1) = CDA, BDA, BCA, BCD, ADA, ACA, ACD, ABA, ABD, ABC = 10
$$
$$
s(3, str1) = DA, CA, CD, BA, BD, BC, AA, AD, AC, AB = 10
$$
$$
s(4, str1) = A, D, C, B, A = 5
$$
and gives 30 in total, and only 29 unique strings. In general, the total combination could be solved using the formula $sum_{k=1}^{r} {{n}choose{k}}$. However I could not find a formula to calculate total unique combination from a given string since the relative position of characters determine the total unique combination. Let's say that another 5-character long string is str2 = {AAABC}, then the number of unique sequences will be different:
$$
s(1, str2) = AABC, AABC, AABC, AAAC, AAAB
$$
$$
s(2, str2) = ABC, ABC, AAC, AAB, ABC, AAC, AAB, AAC, AAB, AAA
$$
$$
s(3, str2) = BCA, AC, AB, AC, AB, AA, AC, AB, AA, AA
$$
$$
s(4, str3) = C, B, A, A, A
$$
The total combinations is still 30, but the unique combinations reduce to 14. Is there any way to count such unique combination for any string with length n composed of four characters (A, B, C, and D)?






combinatorics







share|cite|improve this question









$endgroup$












  • $begingroup$
    Answered on stackoverflow: stackoverflow.com/questions/50702755/…, stackoverflow.com/questions/5151483/….
    $endgroup$
    – Yuval Filmus
    Jan 11 at 9:08










  • $begingroup$
    Thanks for the above link @YuvalFilmus. However, I need the mathematical formula which could be applied to a general problem of string deletion.
    $endgroup$
    – Vic Brown
    Jan 14 at 2:23














1












1








1





$begingroup$


Is there any general formula to determine a unique combination for character deletion in a string comprised of 4 characters? Let's say that I have $str1 = {ABCDA}$ and I want to delete from one up to four position in that string. Thus all the combination for deletion up to four position of str1 would be:
$$
s(1, str1) = BCDA, ACDA, ABDA, ABCA, ABCD = 5
$$
$$
s(2, str1) = CDA, BDA, BCA, BCD, ADA, ACA, ACD, ABA, ABD, ABC = 10
$$
$$
s(3, str1) = DA, CA, CD, BA, BD, BC, AA, AD, AC, AB = 10
$$
$$
s(4, str1) = A, D, C, B, A = 5
$$
and gives 30 in total, and only 29 unique strings. In general, the total combination could be solved using the formula $sum_{k=1}^{r} {{n}choose{k}}$. However I could not find a formula to calculate total unique combination from a given string since the relative position of characters determine the total unique combination. Let's say that another 5-character long string is str2 = {AAABC}, then the number of unique sequences will be different:
$$
s(1, str2) = AABC, AABC, AABC, AAAC, AAAB
$$
$$
s(2, str2) = ABC, ABC, AAC, AAB, ABC, AAC, AAB, AAC, AAB, AAA
$$
$$
s(3, str2) = BCA, AC, AB, AC, AB, AA, AC, AB, AA, AA
$$
$$
s(4, str3) = C, B, A, A, A
$$
The total combinations is still 30, but the unique combinations reduce to 14. Is there any way to count such unique combination for any string with length n composed of four characters (A, B, C, and D)?






combinatorics







share|cite|improve this question









$endgroup$




Is there any general formula to determine a unique combination for character deletion in a string comprised of 4 characters? Let's say that I have $str1 = {ABCDA}$ and I want to delete from one up to four position in that string. Thus all the combination for deletion up to four position of str1 would be:
$$
s(1, str1) = BCDA, ACDA, ABDA, ABCA, ABCD = 5
$$
$$
s(2, str1) = CDA, BDA, BCA, BCD, ADA, ACA, ACD, ABA, ABD, ABC = 10
$$
$$
s(3, str1) = DA, CA, CD, BA, BD, BC, AA, AD, AC, AB = 10
$$
$$
s(4, str1) = A, D, C, B, A = 5
$$
and gives 30 in total, and only 29 unique strings. In general, the total combination could be solved using the formula $sum_{k=1}^{r} {{n}choose{k}}$. However I could not find a formula to calculate total unique combination from a given string since the relative position of characters determine the total unique combination. Let's say that another 5-character long string is str2 = {AAABC}, then the number of unique sequences will be different:
$$
s(1, str2) = AABC, AABC, AABC, AAAC, AAAB
$$
$$
s(2, str2) = ABC, ABC, AAC, AAB, ABC, AAC, AAB, AAC, AAB, AAA
$$
$$
s(3, str2) = BCA, AC, AB, AC, AB, AA, AC, AB, AA, AA
$$
$$
s(4, str3) = C, B, A, A, A
$$
The total combinations is still 30, but the unique combinations reduce to 14. Is there any way to count such unique combination for any string with length n composed of four characters (A, B, C, and D)?






combinatorics




combinatorics






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share|cite|improve this question











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asked Jan 10 at 7:21









Vic BrownVic Brown

213




213












  • $begingroup$
    Answered on stackoverflow: stackoverflow.com/questions/50702755/…, stackoverflow.com/questions/5151483/….
    $endgroup$
    – Yuval Filmus
    Jan 11 at 9:08










  • $begingroup$
    Thanks for the above link @YuvalFilmus. However, I need the mathematical formula which could be applied to a general problem of string deletion.
    $endgroup$
    – Vic Brown
    Jan 14 at 2:23


















  • $begingroup$
    Answered on stackoverflow: stackoverflow.com/questions/50702755/…, stackoverflow.com/questions/5151483/….
    $endgroup$
    – Yuval Filmus
    Jan 11 at 9:08










  • $begingroup$
    Thanks for the above link @YuvalFilmus. However, I need the mathematical formula which could be applied to a general problem of string deletion.
    $endgroup$
    – Vic Brown
    Jan 14 at 2:23
















$begingroup$
Answered on stackoverflow: stackoverflow.com/questions/50702755/…, stackoverflow.com/questions/5151483/….
$endgroup$
– Yuval Filmus
Jan 11 at 9:08




$begingroup$
Answered on stackoverflow: stackoverflow.com/questions/50702755/…, stackoverflow.com/questions/5151483/….
$endgroup$
– Yuval Filmus
Jan 11 at 9:08












$begingroup$
Thanks for the above link @YuvalFilmus. However, I need the mathematical formula which could be applied to a general problem of string deletion.
$endgroup$
– Vic Brown
Jan 14 at 2:23




$begingroup$
Thanks for the above link @YuvalFilmus. However, I need the mathematical formula which could be applied to a general problem of string deletion.
$endgroup$
– Vic Brown
Jan 14 at 2:23










1 Answer
1






active

oldest

votes


















0












$begingroup$

I found a hint for calculating unique combination for one character deletion such that the unique combination is equal to total combination as long as there is no consecutive character repeat:
$$
s(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
$$
$$
s_{U}(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
$$
When repeats of characters occur, then the unique combination will be reduced:
$$
s(1, AABCD) = ABCD, ABCD, AACD, AABD, AABC
$$
$$
s_{U}(1, AABCD) = ABCD, AABD, AACD, AABC
$$
from this observation I conclude that the unique combination of any string of length n depends on the occurrence of runs, such that:
$$
s_{U}(1, AABCD) = s(1, AABCD) - (rtimes d)
$$
$$
= {{5}choose{1}} - (1 . 1) = 4
$$
I find it also true for string containing multiple runs, such as AAABBCD, where: $s_{U} = {{7}choose{1}} - ((1times 2)+(1times 1)) = 4$. However I find difficulties when I try the same formula for 2, 3, or four character deletion. My hypothesis is that the unique combination can be obtained by subtracting the total combination with some factors. I presumed that these factors involve the occurrence of repeats.



Any help for this problem?






share|cite|improve this answer









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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

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    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    I found a hint for calculating unique combination for one character deletion such that the unique combination is equal to total combination as long as there is no consecutive character repeat:
    $$
    s(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
    $$
    $$
    s_{U}(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
    $$
    When repeats of characters occur, then the unique combination will be reduced:
    $$
    s(1, AABCD) = ABCD, ABCD, AACD, AABD, AABC
    $$
    $$
    s_{U}(1, AABCD) = ABCD, AABD, AACD, AABC
    $$
    from this observation I conclude that the unique combination of any string of length n depends on the occurrence of runs, such that:
    $$
    s_{U}(1, AABCD) = s(1, AABCD) - (rtimes d)
    $$
    $$
    = {{5}choose{1}} - (1 . 1) = 4
    $$
    I find it also true for string containing multiple runs, such as AAABBCD, where: $s_{U} = {{7}choose{1}} - ((1times 2)+(1times 1)) = 4$. However I find difficulties when I try the same formula for 2, 3, or four character deletion. My hypothesis is that the unique combination can be obtained by subtracting the total combination with some factors. I presumed that these factors involve the occurrence of repeats.



    Any help for this problem?






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      I found a hint for calculating unique combination for one character deletion such that the unique combination is equal to total combination as long as there is no consecutive character repeat:
      $$
      s(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
      $$
      $$
      s_{U}(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
      $$
      When repeats of characters occur, then the unique combination will be reduced:
      $$
      s(1, AABCD) = ABCD, ABCD, AACD, AABD, AABC
      $$
      $$
      s_{U}(1, AABCD) = ABCD, AABD, AACD, AABC
      $$
      from this observation I conclude that the unique combination of any string of length n depends on the occurrence of runs, such that:
      $$
      s_{U}(1, AABCD) = s(1, AABCD) - (rtimes d)
      $$
      $$
      = {{5}choose{1}} - (1 . 1) = 4
      $$
      I find it also true for string containing multiple runs, such as AAABBCD, where: $s_{U} = {{7}choose{1}} - ((1times 2)+(1times 1)) = 4$. However I find difficulties when I try the same formula for 2, 3, or four character deletion. My hypothesis is that the unique combination can be obtained by subtracting the total combination with some factors. I presumed that these factors involve the occurrence of repeats.



      Any help for this problem?






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        I found a hint for calculating unique combination for one character deletion such that the unique combination is equal to total combination as long as there is no consecutive character repeat:
        $$
        s(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
        $$
        $$
        s_{U}(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
        $$
        When repeats of characters occur, then the unique combination will be reduced:
        $$
        s(1, AABCD) = ABCD, ABCD, AACD, AABD, AABC
        $$
        $$
        s_{U}(1, AABCD) = ABCD, AABD, AACD, AABC
        $$
        from this observation I conclude that the unique combination of any string of length n depends on the occurrence of runs, such that:
        $$
        s_{U}(1, AABCD) = s(1, AABCD) - (rtimes d)
        $$
        $$
        = {{5}choose{1}} - (1 . 1) = 4
        $$
        I find it also true for string containing multiple runs, such as AAABBCD, where: $s_{U} = {{7}choose{1}} - ((1times 2)+(1times 1)) = 4$. However I find difficulties when I try the same formula for 2, 3, or four character deletion. My hypothesis is that the unique combination can be obtained by subtracting the total combination with some factors. I presumed that these factors involve the occurrence of repeats.



        Any help for this problem?






        share|cite|improve this answer









        $endgroup$



        I found a hint for calculating unique combination for one character deletion such that the unique combination is equal to total combination as long as there is no consecutive character repeat:
        $$
        s(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
        $$
        $$
        s_{U}(1, ABCDA) = BCDA, ACDA, ABDA, ABCA, ABCD
        $$
        When repeats of characters occur, then the unique combination will be reduced:
        $$
        s(1, AABCD) = ABCD, ABCD, AACD, AABD, AABC
        $$
        $$
        s_{U}(1, AABCD) = ABCD, AABD, AACD, AABC
        $$
        from this observation I conclude that the unique combination of any string of length n depends on the occurrence of runs, such that:
        $$
        s_{U}(1, AABCD) = s(1, AABCD) - (rtimes d)
        $$
        $$
        = {{5}choose{1}} - (1 . 1) = 4
        $$
        I find it also true for string containing multiple runs, such as AAABBCD, where: $s_{U} = {{7}choose{1}} - ((1times 2)+(1times 1)) = 4$. However I find difficulties when I try the same formula for 2, 3, or four character deletion. My hypothesis is that the unique combination can be obtained by subtracting the total combination with some factors. I presumed that these factors involve the occurrence of repeats.



        Any help for this problem?







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 14 at 3:02









        Vic BrownVic Brown

        213




        213






























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