In need of interpretation of a ordered-subset related problem.












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$begingroup$


''Given a set $S_n = {1, 2, 3, ldots, n}$, we define a preference list to be an ordered subset of $S_n$. Let $P_n$ be the number of preference lists of $S_n$. Show that for positive integers $n > m$, $P_n - P_m$ is divisible by $n - m$.



Note: the empty set and $S_n$ are subsets of $S_n$.''



Can anyone help me out to make me understand the problem ? (giving an explained example of this question will be very good )
Thanks in advance :)










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    0












    $begingroup$


    ''Given a set $S_n = {1, 2, 3, ldots, n}$, we define a preference list to be an ordered subset of $S_n$. Let $P_n$ be the number of preference lists of $S_n$. Show that for positive integers $n > m$, $P_n - P_m$ is divisible by $n - m$.



    Note: the empty set and $S_n$ are subsets of $S_n$.''



    Can anyone help me out to make me understand the problem ? (giving an explained example of this question will be very good )
    Thanks in advance :)










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      ''Given a set $S_n = {1, 2, 3, ldots, n}$, we define a preference list to be an ordered subset of $S_n$. Let $P_n$ be the number of preference lists of $S_n$. Show that for positive integers $n > m$, $P_n - P_m$ is divisible by $n - m$.



      Note: the empty set and $S_n$ are subsets of $S_n$.''



      Can anyone help me out to make me understand the problem ? (giving an explained example of this question will be very good )
      Thanks in advance :)










      share|cite|improve this question









      $endgroup$




      ''Given a set $S_n = {1, 2, 3, ldots, n}$, we define a preference list to be an ordered subset of $S_n$. Let $P_n$ be the number of preference lists of $S_n$. Show that for positive integers $n > m$, $P_n - P_m$ is divisible by $n - m$.



      Note: the empty set and $S_n$ are subsets of $S_n$.''



      Can anyone help me out to make me understand the problem ? (giving an explained example of this question will be very good )
      Thanks in advance :)







      combinatorics number-theory






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      asked Dec 29 '18 at 13:10









      nahin munkarnahin munkar

      1




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          1 Answer
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          $begingroup$

          For example, for $n=3$ the preference lists of ${1,2,3}$ are
          $$ emptyset\
          (1)\
          (2)\
          (3)\
          (1, 2)\
          (2, 1)\
          (1, 3)\
          (3, 1)\
          (2, 3)\
          (3, 2)\
          (1, 2, 3)\
          (1, 3, 2)\
          (2, 1, 3)\
          (2, 3, 1)\
          (3, 1, 2)\
          (3, 2, 1)$$






          share|cite|improve this answer









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            1 Answer
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            active

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            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            0












            $begingroup$

            For example, for $n=3$ the preference lists of ${1,2,3}$ are
            $$ emptyset\
            (1)\
            (2)\
            (3)\
            (1, 2)\
            (2, 1)\
            (1, 3)\
            (3, 1)\
            (2, 3)\
            (3, 2)\
            (1, 2, 3)\
            (1, 3, 2)\
            (2, 1, 3)\
            (2, 3, 1)\
            (3, 1, 2)\
            (3, 2, 1)$$






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              For example, for $n=3$ the preference lists of ${1,2,3}$ are
              $$ emptyset\
              (1)\
              (2)\
              (3)\
              (1, 2)\
              (2, 1)\
              (1, 3)\
              (3, 1)\
              (2, 3)\
              (3, 2)\
              (1, 2, 3)\
              (1, 3, 2)\
              (2, 1, 3)\
              (2, 3, 1)\
              (3, 1, 2)\
              (3, 2, 1)$$






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                For example, for $n=3$ the preference lists of ${1,2,3}$ are
                $$ emptyset\
                (1)\
                (2)\
                (3)\
                (1, 2)\
                (2, 1)\
                (1, 3)\
                (3, 1)\
                (2, 3)\
                (3, 2)\
                (1, 2, 3)\
                (1, 3, 2)\
                (2, 1, 3)\
                (2, 3, 1)\
                (3, 1, 2)\
                (3, 2, 1)$$






                share|cite|improve this answer









                $endgroup$



                For example, for $n=3$ the preference lists of ${1,2,3}$ are
                $$ emptyset\
                (1)\
                (2)\
                (3)\
                (1, 2)\
                (2, 1)\
                (1, 3)\
                (3, 1)\
                (2, 3)\
                (3, 2)\
                (1, 2, 3)\
                (1, 3, 2)\
                (2, 1, 3)\
                (2, 3, 1)\
                (3, 1, 2)\
                (3, 2, 1)$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 29 '18 at 14:13









                awkwardawkward

                6,92511026




                6,92511026






























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