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$n++ ;
$tmp = $tmp . $_  if $n < 11  ; 
print  $_ . $tmp  if $n == 11 ;
print             if $n > 11  ;

$n++ ;
$tmp = $tmp . $_  if $n < 11  ; 
print  $_ . $tmp  if $n == 11 ;
print  $_         if $n > 11  ;

AWK (stolen from Ed (not the editor!))

NR < 11 { 
    buf[NR] = $0; next 
}
NR >=11 {
    print
    if (NR == 11) {
        for (i=1; i<11; i++) { print buf[i] }
    }
}

I used NR instead of incrementing n, and made the flow more explicit. Same "trick": the next simplifies downstream.

With perl -n

$n++ ;
$tmp = $tmp . $_  if $n < 11  ; 
print  $_ . $tmp  if $n == 11 ;
print             if $n > 11  ;

This is the best format. Symmetrical.

And as some script:

$lnum=11 ;
while (<>) {
    $n++ ;
    if ($n < $lnum) {  $sto .= $_  }
    else {                                            # lnum is reached
        if ($n == $lnum) { print $_ . $sto  }         # append stored lines
         else             { print $_  }                # just print (neither <lnum nor ==lnum)
    }
}

An if less but a nested else more.

And as some script, which takes a file name and needs no option:

$want=11 ;
while (<>) {
    $n++ ;
    if ($n < $want)            # before $want: store line
        { $lowlines .= $_ ;  
          next }               # next line (avoids 'else')   
    if ($n == $want)           # at line $want: append stored lines to $_     
        { $_ .= $lowlines }   
    print ;                    # print $_ for $n not less than $want
}