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Inverse of a particular bijection

1

1

$begingroup$

Let $X := (i,j) in mathbbNtimesmathbbN ; $. I know that the function $T: Xlongrightarrow mathbbN$ defined by $$T(i,j) = frac 12 j(j-3) + i + 1$$ is a bijection. I am interested in the inverse of $T$. Is it possible to find an explicit formula for the inverse of $T$? Any help or comment would be helpful.

combinatorics number-theory discrete-mathematics elementary-set-theory

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edited Mar 31 at 15:50

Maria Mazur

49.9k1361125

asked Mar 31 at 14:07

Sara.TSara.T

300110

$endgroup$

add a comment | 

1

1

$begingroup$

Let $X := (i,j) in mathbbNtimesmathbbN ; $. I know that the function $T: Xlongrightarrow mathbbN$ defined by $$T(i,j) = frac 12 j(j-3) + i + 1$$ is a bijection. I am interested in the inverse of $T$. Is it possible to find an explicit formula for the inverse of $T$? Any help or comment would be helpful.

combinatorics number-theory discrete-mathematics elementary-set-theory

share|cite|improve this question

edited Mar 31 at 15:50

Maria Mazur

49.9k1361125

asked Mar 31 at 14:07

Sara.TSara.T

300110

$endgroup$

add a comment | 

$begingroup$

Let $X := (i,j) in mathbbNtimesmathbbN ; $. I know that the function $T: Xlongrightarrow mathbbN$ defined by $$T(i,j) = frac 12 j(j-3) + i + 1$$ is a bijection. I am interested in the inverse of $T$. Is it possible to find an explicit formula for the inverse of $T$? Any help or comment would be helpful.

combinatorics number-theory discrete-mathematics elementary-set-theory

share|cite|improve this question

edited Mar 31 at 15:50

Maria Mazur

49.9k1361125

asked Mar 31 at 14:07

Sara.TSara.T

300110

$endgroup$

share|cite|improve this question

edited Mar 31 at 15:50

Maria Mazur

49.9k1361125

asked Mar 31 at 14:07

Sara.TSara.T

300110

share|cite|improve this question

edited Mar 31 at 15:50

Maria Mazur

49.9k1361125

asked Mar 31 at 14:07

Sara.TSara.T

300110

edited Mar 31 at 15:50

Maria Mazur

49.9k1361125

edited Mar 31 at 15:50

Maria Mazur

49.9k1361125

asked Mar 31 at 14:07

Sara.TSara.T

300110

asked Mar 31 at 14:07

Sara.TSara.T

300110

1 Answer
1

active

oldest

votes

1

$begingroup$

Let $nin mathbbN$ and let $j$ be the first positive integer such that $$nleq jchoose 2$$ and let $$ i = n- frac 12 j(j-3) -1$$

then the map $$n mapsto (i,j)$$ is the inverse of the given map.

---

Actually $$j = Big[sqrt8n+1+1over 2Big], $$

where $[x]$ is the first integer not smaller than $x$.

share|cite|improve this answer

edited Mar 31 at 14:56

J. W. Tanner

4,7721420

answered Mar 31 at 14:15

Maria MazurMaria Mazur

49.9k1361125

$endgroup$

add a comment | 

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1

$begingroup$

Let $nin mathbbN$ and let $j$ be the first positive integer such that $$nleq jchoose 2$$ and let $$ i = n- frac 12 j(j-3) -1$$

then the map $$n mapsto (i,j)$$ is the inverse of the given map.

---

Actually $$j = Big[sqrt8n+1+1over 2Big], $$

where $[x]$ is the first integer not smaller than $x$.

share|cite|improve this answer

edited Mar 31 at 14:56

J. W. Tanner

4,7721420

answered Mar 31 at 14:15

Maria MazurMaria Mazur

49.9k1361125

$endgroup$

add a comment | 

1

$begingroup$

Let $nin mathbbN$ and let $j$ be the first positive integer such that $$nleq jchoose 2$$ and let $$ i = n- frac 12 j(j-3) -1$$

then the map $$n mapsto (i,j)$$ is the inverse of the given map.

---

Actually $$j = Big[sqrt8n+1+1over 2Big], $$

where $[x]$ is the first integer not smaller than $x$.

share|cite|improve this answer

edited Mar 31 at 14:56

J. W. Tanner

4,7721420

answered Mar 31 at 14:15

Maria MazurMaria Mazur

49.9k1361125

$endgroup$

add a comment | 

$begingroup$

Let $nin mathbbN$ and let $j$ be the first positive integer such that $$nleq jchoose 2$$ and let $$ i = n- frac 12 j(j-3) -1$$

then the map $$n mapsto (i,j)$$ is the inverse of the given map.

---

Actually $$j = Big[sqrt8n+1+1over 2Big], $$

where $[x]$ is the first integer not smaller than $x$.

share|cite|improve this answer

edited Mar 31 at 14:56

J. W. Tanner

4,7721420

answered Mar 31 at 14:15

Maria MazurMaria Mazur

49.9k1361125

$endgroup$

share|cite|improve this answer

edited Mar 31 at 14:56

J. W. Tanner

4,7721420

answered Mar 31 at 14:15

Maria MazurMaria Mazur

49.9k1361125

edited Mar 31 at 14:56

J. W. Tanner

4,7721420

edited Mar 31 at 14:56

J. W. Tanner

4,7721420

answered Mar 31 at 14:15

Maria MazurMaria Mazur

49.9k1361125

answered Mar 31 at 14:15

Maria MazurMaria Mazur

49.9k1361125