Japt, 8 bytes

ô ¤¬ñ i1

Try it

ô ¤¬ñ i1     :Implicit input of integer
ô            :Range [0,input]
  ¤          :Slice off the first 2 elements
   ¬         :Join
    ñ        :Sort
      i1     :Prepend 1

Husk, 8 bytes

:'1Oṁs…2

Try it online!

Corrected my previous answer.

JavaScript ES6, 49 46 Bytes

Thanks to edc65 for 2 bytes

F=
x=>1+[...(g=_=>x>1?x--+g``:_)``].sort().join``

;console.log(F(15))

C (gcc), 172 161 bytes

• Saved eleven bytes thanks to ceilingcat.

f(n){char s[++n];int d[10]={0},j=1,k;for(;sprintf(s,"%d",j),j++<n;)for(k=0;s[k];)d[s[k++]-48]++;d[printf("1")]--;for(j=0;j<10;j++)for(k=d[j];k--;putchar(j+48));}

Try it online!

Attache, 20 bytes

{"1"+Sort!Join[2:_]}

Try it online!

The point-free version is actually 1 byte longer: "1"&`+@Sort@Join@2&`:. This works as most of the other answers do.

Stacked, 22 bytes

2|>''join sorted'1'+

Try it here! Expects input on TOS. Example on the repl:

> 15
(15)
> 2|>''join sorted'1'+
('101111111223344556789')

Explanation

2|>''join sorted'1'+
2|>                    range from 2 to n
    ''join              join by empty string
           sorted       sort it
                 '1'+  and prepend a '1'

A more fun version:

Stacked, 45 bytes

{*x:1[1+:yield]x 1-*}!toarr''join sorted'1'+

Expects input as TOS. For example (on the repl):

> 15    (* the input *)
(15)
> {*x:1[1+:yield]x 1-*}!toarr''join sorted'1'+
('101111111223344556789')
> 

Explanation

{*x:1[1+:yield]x 1-*}!toarr''join sorted'1'+
{*x:1[1+:yield]x 1-*}                            generator function that...
  x:                                               given x
    1                                              pushes 1
     [        ]x 1-*                               and executes the inside (x-1) times:
      1+                                             adds 1 to the TOS (initially 1);
                                                     2 is the first value yielded
        :yield                                       and yields it for the generator
                     !                           initialize this generator with TOS
                      toarr                      exhaust generator into an array
                           ''join                join by empty strings
                                  sorted         sort the values
                                        '1'+    prepend '1'.

Lua, 102 Bytes

How can lua be so long while all other entries are < 30 bytes... I'm sure even Java would be shorter!

t={}for i=2,...do(i..''):gsub(".",function(d)t[#t+1]=d end)end
table.sort(t)print(1 ..table.concat(t))

Ungolfed

t={}                           -- initialise t as a table
for i=2,...                    -- iterate from 2 to n
do
  (i..''):gsub(".",function(d) -- iterate over each digits in i
    t[#t+1]=d                  -- insert the digit into t
  end)
end
table.sort(t)                 -- sort t
print(1 ..table.concat(t))    -- prepend a 1 to the content of t separated by ''

I'm forced to put a space in print(1 .., else it would try to evaluate it to a decimal 1.<decimal part>, and throws an error because . isn't a valid decimal part.

Oracle SQL 11.2, 222 211 bytes

SELECT 1||TRIM(xmlagg(xmlelement(e,c)).EXTRACT('//text()'))FROM(SELECT SUBSTR(s,LEVEL+2,1)c FROM(SELECT MAX(sys_connect_by_path(LEVEL,' '))s FROM DUAL CONNECT BY LEVEL<=:1)CONNECT BY LEVEL<=LENGTH(s)ORDER BY 1);

Un-golfed

SELECT 1||TRIM(xmlagg(xmlelement(e,c)).EXTRACT('//text()'))  -- add leading 1, concatenate each char and remove spaces
FROM   (
         SELECT SUBSTR(s,LEVEL+2,1)c                          -- split the string in characters, omiting the first number (1)   
         FROM   (
                  SELECT MAX(sys_connect_by_path(LEVEL,' '))s -- create a string by concatenating numbers
                  FROM   DUAL 
                  CONNECT BY LEVEL<=:1
                )
         CONNECT BY LEVEL<=LENGTH(s)ORDER BY 1
       )

Pyke, 9 bytes, noncompetitive

Stm`sS1R+

Explanation:

S         -      range(1,inp+1)
 t        -     ^[1:]
  m`      -    map(str,^)
    s     -   sum(^)
     S    -  sort(^)
      1R+ - 1+^

Try it here!

Seriously, 10 bytes

,Rp@εjS@$+

Try it online!

Explanation:

,Rp@εjS@$+
,Rp@        push range(1, input()+1), pop first element (1), swap
    εjS     join list on empty string, sort
       @$+  prepend the 1

Reng v.3.2, 36 bytes

1i#x:xe!Ø;$:1+)å(Eh
å'11µ1nran! ¡n~

Try it here!

This pushes 1 to the stack, then stores the input i into x (i#x). Then, a range of all the necessary numbers is created using :xe!Ø;$:1+)å(Eh, being ended by Ø, which goes the first i.e. next line. å takes every number and pushes the digits of that number.

On the next line, å breaks the last number into its digits, ' sorts the stack, and 11µ removes 1 1 from the stack. A 1 is then printed, and the numbers are reversed then outputted through the loop an! ¡n. The program is terminated with ~.

Jelly, 8 bytes

RDFṢ1œ|Ḍ

Try it online!

How it works

RDFṢ1œ|Ḍ  Main link. Argument: n (integer)

R         Range; yield r := [1, ..., n].
 D        Convert each k in r into the array of its digits in base 10.
  F       Flatten the resulting array of lists.
   Ṣ      Sort the resulting flat list of digits.
    1œ|   Perform multiset union with 1 as left argument.
          This moves a single 1 to the beginning of the list.
       Ḍ  Convert the resulting list of base 10 to integer.

Mathcad, 86 "bytes"

The function s(n) uses a for loop to build up the Champernowne "integer" by converting each number to its string form and concatenating them together. The string is then converted to its equivalent vector of ASCII codes and sorted. The function then swaps any leading zeros with the first one, finally converting the vector back to a string.

To check the function, I put the test cases into a vector vn, then applied s to vn using the vectorize operator. I then check the results against the given test case values.

Mathcad is mathematical application based on 2D worksheets comprised of "regions" each of which can be text, a mathematical expression, program, plot or scripted component.

A mathematical or programming instruction is picked from a palette toolbar or entered using a keyboard shortcut. For golfing purposes, an operation ("byte") is taken to be the number of keyboard operations necessary to create a name or expression (for example, to set the variable a to 3, we would write a:=3. The definition operator := is a single keypress ":", as are a and 3 giving a total of 3 "bytes". The programming for operator requires typing ctl-shft-# (or a single click on the programming toolbar) so again is equivalent to 1 byte.

Gogh, 9 7 bytes

GJT1-1P

You can run this using:

$ ./gogh noi 'GJT1-1P' <input>

G     “ Push a range (1, TOS]       ”
J     “ Join the TOS                ”
T     “ Sort the TOS                ”
1-    “ Remove the first 1          ”
P     “ Prepend the TOS to the STOS ”

05AB1E, 6 bytes

Note: This submission uses features that postdate this challenge and is therefore not competitive

Code:

Lß?J{?

Explanation:

L      # Creates the list [1 .. input]
 ß     # Extract the smallest value of the list
  ?    # Print this value (always 1)
   J   # ''.join(list)
    {  # Sort the string
     ? # Print this value

Uses ISO 8859-1 encoding

Bash, 35 34 bytes

printf %d 1`seq 2 $1|fold -1|sort`

This builds on the answers of @DigitalTrauma and @OlivierDulac. Try it online with Ideone.

How it works

• seq 2 $1 prints all integers from 2 to the one specified on the command line.

• fold -1 wraps all lines with width 1, i.e., places each character on its own line.

  -1 seems to be an undocumented feature.

• sort sorts the characters by their numeric value.

• printf %d 1`...`​ prepends a 1 to the first line and prints each line as an integer (%d), with no separation.

  This takes advantage of Bash's curious printf implementation, which repeats the format string over and over, until all arguments are consumed.

Bash and GNU tools, 58 52 bytes

echo 1$(seq 2 $1|sed -e 's/./&n/g'|sort|tr -d \n)

Brachylog, 76 41 bytes

1 .;{,1:.e?}?:1fcbZlL,ZoOlM,10^(L-M)=:Oc.

Takes a number as input.

This solution works with the few changes I made to the built-in Findall predicate f. OP is apparently OK with using languages older than the answer so I think this is fine (the changes I made were intented a long time ago, I just motivated myself to do it because of this challenge).

Explanation

1 .                                            § If the input is 1, unify output with 1

   ;                                           § Else

    {      }?:1f                               § Output a list of all inputs which satisfy
                                               § the predicate in brackets with the input
                                               § of the main predicate (ie the input number)
                                               § as output

     ,1:.e?                                    § True if the input is an integer between 1
                                               § and . (the output)

                cbZ                            § Concatenate everything into a single number,
                                               § remove the first digit (1) and call it Z

                   lL,ZoOlM,                   § L is the length of Z, M is the length of O
                                               § O being Z sorted (which removes the leading
                                               § 0s)

                            10^(L-M)=:Oc.      § Concatenate 10^(L-M) at the beginning of O
                                               § and unify it with the output

Seriously, 10 bytes

,u2xεjS'1+

Hex Dump:

2c753278ee6a5327312b

Try It Online

Explanation:

,            Read input
 u2x         Push range from 2..n
    εj       Join into string
      S      Sort
       '1+   Prepend a "1"

MATL, 17 bytes

Uses current version (7.0.0) of language/compiler.

49[]i:"@YUh]6L)Sh

Inspired by FryTheEgggman's answer.

EDIT (July 29, 2016): You can try it online with some modifications to conform to changes in the language.

Example

>> matl
 > 49[]i:"@YUh]6L)Sh
 >
> 12
101111223456789

Explanation

49        % push '1'
[]        % push empty array
i:        % input "N" and build vector [1, 2, ... N]
"         % for each number in this vector
   @      % push that number
   YU     % convert to string
   h      % concatenate horizontally
]         % end
6L)       % remove first element
S         % sort
h         % concatenate horizontally

Mathematica, 52 bytes

"1"<>ToString/@Sort[Join@@IntegerDigits[2~Range~#]]&

Once again, string processing happened...

Milky Way 1.6.4, 22 bytes (not competing)

^^'LH=^^JB", "-Q"1";+!

I had to add a sorting opcode for this challenge.

Explanation

^^                      ` pop the TOS
  '                     ` read input from the command line
   LH                   ` push a reversed Pythonic range(TOS+1)
     =^^J               ` remove the top 2 items from the TOS
         B              ` push the string literal of the TOS
          ", "-         ` remove ", " from the TOS
               Q        ` sort the TOS
                "1";+   ` add "1" to the beginning of the TOS
                     !  ` output the TOS

Ruby, 48 bytes

An anonymous function. Basically just Rubyfied some of the other answers here..

->n{n>1?(?1+[*2..n].join.chars.sort*'').to_i: n}

APL (17)

'1',∆[⍋∆←1↓∊⍕¨⍳⎕]

Explanation:

'1',∆[⍋∆←1↓∊⍕¨⍳⎕]
-----------------
               ⎕   read a number from the keyboard
               ⍳    get the natural numbers up to and including that number
             ⍕¨    get the string representation for each number
           ∊       flatten the array (giving a string of digits)
         1↓        remove the first digit (which is always 1)
       ∆←          store the result in ∆
      ⍋            get a permutation to sort ∆ upwards
    ∆[           ] rearrange ∆ so that it is sorted
'1',               add a 1 to the front

Test:

      '1',∆[⍋∆←1↓∊⍕¨⍳⎕]
⎕:
      1
1
      '1',∆[⍋∆←1↓∊⍕¨⍳⎕]
⎕:
      10
10123456789
      '1',∆[⍋∆←1↓∊⍕¨⍳⎕]
⎕:
      55
10000011111111111111122222222222222223333333333333333444444444444444455555555555566666777778888899999

Smalltalk, 76 bytes

As usual in Smalltalk, conceptually very terse, but textually very verbose...

f:l^'1',((2to:l)fold:[:p :q|p asString,q asString])asByteArray sort asString

Add this as a class method for String and call like this, e.g. for 20, String f: 20

ClojureScript, 48 bytes

#(apply str"1"(sort(apply str(range 2(inc %)))))

Same as all of the others, pretty much. REPL available here.

Python 2, 60 bytes

P=input();print"1"+"".join(sorted(`range(2,P+1)`)[P*2-4:-2])

Indexing for the win. :-)

Python 2, 60 bytes

P=input();print"1"+"".join(sorted(`range(-P,-1)`))[P*3-5:-2]

Just an alternative.

Perl, 44 42 41 33 31 bytes

Yaaay, first post ever !

Thanks to primo for the 2 bytes save.

print 1,sort"@{[2..<>]}"=~/d/g

As others did, removing 1 and prepending it manually does the work.

Try it online

APL, 21 bytes

{' '~⍨⍕1,x[⍋x←⍕1↓⍳⍵]}

This is an unnamed monadic function that accepts an integer on the right and returns a string. To call it, assign it to a variable.

Explanation:

            x←⍕1↓⍳⍵]}   ⍝ Get the numbers 1:input, drop 1, convert to string
         x[⍋            ⍝ Sort
      ⍕1,               ⍝ Tack 1 onto the front, flatten to string
{' '~⍨                  ⍝ Remove the spaces (side effect of ⍕ on an array)

Try it online

Python 2, 60 bytes

_="".join;print"1"+_(sorted(_(map(str,range(2,input()+1)))))

CJam, 9

1qi),2>s$

Try it online

Alternatively:

qi,:)(s$

PowerShell, 61 59 bytes

param($a)(("1"+-join([char[]]-join(2..$a)|sort)),1)[$a-eq1]

Takes input param($a) and then uses it to index into an array with [$a-eq1]. If true, we index the second element and output 1. Otherwise, we concatenate "1" with the joined array created by 1) defining a new range 2..$a that has been itself joined together, 2) casting that as a char-array, and 3) sending it through the Sort-Object cmdlet, all of which is then output.

_{Edit1 -- Saved 2 bytes by moving the inner -join operator.}

Julia, 33 bytes

n->"1"*join(sort([join(2:n)...]))

This is a lambda function that accepts an integer and returns a string. To call it, assign it to a variable.

We construct the range 2:n, which will be empty for n < 2, join it into a string, splat the string into an array of characters, sort them, join it into a string, and prepend 1.

Retina, 78 bytes

Time to show off some new Retina features (it's still not very competitive, but before today this would probably have been closer to 300 bytes).

.+
$0$*1
B
 $`
(1)+
$#1
^1| 

.
 1$0$*1
+r`(1+2) (1+)b
$2 $1
 1(1)*
$#1
^
1

Try it online.

Explanation

While it's possible to convert between decimal and unary quite conveniently now, this is still quite long because I have to convert back and forth several times because some operations are more doable in decimal than in unary and vice-versa.

.+
$0$*1

Let's start by converting the input to unary. This works by matching the input and then using $*1 which repeats 1 that many times (this repetition feature is new as of today).

B
 $`

Next, we generate a range from 1 to N in unary. I've explained why this works in my FizzBuzz answer.

(1)+
$#1

We convert each number in the range back to decimal so we can work with the decimal digits. This is done by matching each of the unary numbers such that each 1 generates a separate capture. Then we replace that with the number of captures of group one, using the new capture count syntax $#1.

^1| 

This removes the leading 1 as well as all spaces from the string so we're left with only the digits (except for a single 1).

.
 1$0$*1

We convert back to unary and add 1 to each digit (to ensure that even 0 is a non-empty). We also insert a space in front of each digit to ensure that they are separated.

+r`(1+2) (1+)b
$2 $1

We repeatedly match a small number preceded by a larger number and swap them. That's bubble sort in Retina. :)

 1(1)*
$#1

Aaaand back to decimal.

^
1

Finally, we insert a single 1 at the front to account for the one we've removed earlier.

JavaScript (ES6), 65 62 54 52 bytes

Saved 3 bytes thanks to edc65

x=>eval("for(b='';x>1;)1+[...b+=x--].sort().join``")

Builds a string of all numbers from 2 to x, then splits, sorts, joins, and adds a 1 to the beginning. This may still be golfable; suggestions welcome!

Bash + GNU utilities, 58

seq $1|sed 's/./&n/g'|sort|tr -d \n|sed 's/(0*)1/11/'

Try it online.

Haskell, 44 bytes

import Data.List
f n='1':sort(show=<<[2..n])

Unfortunately sort is in Data.List, that's 17 bytes!

Japt, 14 12 bytes

1+2o°U ¬¬n ¬

Try it online!

How it works

1+2o°U ¬¬n ¬  // Implicit: U = input integer
  2o°U        // Generate the range of integers from 2 to U, inclusive.
       ¬¬     // Join, then split into chars.
         n    // Sort.
1+         ¬  // Join again, and add a 1 to the beginning.
              // Implicit: output last expression