public class Category
        {
            public int id { get; set; }
            public string slug { get; set; }
            public string title { get; set; }
            public string description { get; set; }
            public int parent { get; set; }
            public int post_count { get; set; }
        }

List<Category> cats = new List<Category>();

int[] orderData = new[] {3, 17, 13, 176, 4, 177, 179, 178, 180, 181, 182, 183, 184, 185, 186, 139};
List<Category> orderedCategories = cats.OrderBy(item => Array.IndexOf(orderData, item.id)).ToList();

3, 17, 13, 176, 4, 177, 179, 178, 180, 181, 182, 183, 184, 185, 186, 139

public class CategorySortOrder
{
    int CategoryId { get;set;}
    int SortOrder { get;set;}
}

var catSortOrder = new[] {
   new CategorySortOrder() { CategoryId = 3, SortOrder = 1 },
   new CategorySortOrder() { CategoryId = 17, SortOrder = 2 }
   // etc.
   }

var catList = from cat in cats
              join so in catSortOrder on cat.id = so.CategoryId
              select new {
                cat = cat,
                sort_order = so.SortOrder
              };
var finalist = catList
                  .OrderBy(item => item.SortOrder)
                  .Select(item => item.cat)
                  .ToList();

var catList = cats.OrderBy(cat => cat.id).ToList();

var sortedCats=cats.OrderBy(x => x.id).ToList();

public class CategoryComparer : IComparer<Category>
{
    List<int> orderNums =
        new List<int> { 3, 17, 13, 176, 4, 177, 179, 178, 180, 181, 182, 183, 184, 185, 186, 139 };

    public int Compare(Category x, Category y)
    {
        var index1 = orderNums.IndexOf(x.id);
        var index2 = orderNums.IndexOf(y.id);

        if (index1 == index2)
            return 0;

        return index1 > index2 ? 1 : -1;
    }
}

cats.Sort(new CategoryComparer());

var sortedCats = cats.OrderBy(x => x, new CategoryComparer());