Row_number over (Partition by xxx) in Linq?
id |   inst   |   name
  ____________________
  1 | guitar   |  john
  2 |  guitar  |  george
  3 |  guitar  |  paul
  4 |  drums   |  ringo
  5 |  drums   |  pete

IEnumerable <beatle>...

class Beatle
{
  int id;
  string inst;
  string name ;
}

select *,Row_Number() over (partition by inst order by id) as rn from myTalbe

inst    |   name  | rn
_____________________
guitar  |  john   | 1
guitar  |  george | 2
guitar  |  paul   | 3
drums   |  ringo  | 1
drums   |  pete   | 2

var o = beatles.OrderBy(x => x.id).GroupBy(x => x.inst)
.Select(g => new {g, count= g.Count()})
.SelectMany(t => t.g.Select(b => b)
.Zip(Enumerable.Range(1,t.count), (j,i) => new {j.inst, j.name, rn = i}));

foreach (var i in o)
{
    Console.WriteLine("{0} {1} {2}", i.inst, i.name, i.rn);
}

Guitar John 1
Guitar George 2
Guitar Paul 3
drums Ringo 1
drums Pete 2

var grouped = beatles.OrderBy( x => x.id )
  .ToList()   // required because SelectMany below doesn't evaluate to SQL
  .GroupBy( x => x.inst );
var rns = grouped.ToDictionary( x => x.Key, x => 1 );
var result = grouped
  .SelectMany( x => x.Select(
    y => new { inst = y.inst, name = y.name, rn = rns[y.inst]++ } ) );