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class Program
    {
        static void Main(string[] args)
        {
            List<myobject> list = new List<myobject>();
            list.Add(new myobject { name = "n1", recordNumber = 1 });
            list.Add(new myobject { name = "n2", recordNumber = 2 });
            list.Add(new myobject { name = "n3", recordNumber = 3 });
            list.Add(new myobject { name = "n4", recordNumber = 3 });
            //Generates Row Number on the fly
            var withRowNumbers = list 
                    .Select((x, index) => new 
                            {
                                Name = x.name,
                                RecordNumber = x.recordNumber,
                                RowNumber = index + 1
                            }).ToList();
            //Generates Row Number with Partition by clause
            var withRowNumbersPartitionBy = withRowNumbers
                    .OrderBy(x => x.RowNumber)
                    .GroupBy(x => x.RecordNumber)
                    .Select(g => new { g, count = g.Count() })
                    .SelectMany(t => t.g.Select(b => b)
                    .Zip(Enumerable.Range(1, t.count), (j, i) => new { Rn = i, j.RecordNumber, j.Name}))
                    .Where(i=>i.Rn == 1)
                    .ToList();
            //print the result
            withRowNumbersPartitionBy.ToList().ForEach(i => Console.WriteLine("Name =  {0}   RecordNumber = {1}", i.Name, i.RecordNumber));
            Console.ReadKey();
        }
    }
    class myobject
    {
        public int recordNumber { get; set; }
        public string name { get; set; }
    }
Result:
Name =  n1   RecordNumber = 1
Name =  n2   RecordNumber = 2
Name =  n3   RecordNumber = 3