-- Тупо и прямолинейно - это так. Но если вспомнить азы математики... то можно быстрее.
declare @sales table
(
 rowid int identity,
 client_id NVARCHAR(10) NOT NULL, 
 date_ DATE NOT NULL, 
 articul NVARCHAR(10) NOT NULL, 
 check_num NVARCHAR(10) NOT NULL );

INSERT INTO @sales
(client_id, 
 date_, 
 articul, 
 check_num)
VALUES
(
 'id1', '01.01.2020', 'машинка', 'check1'),
(
 'id1', '01.01.2020', 'кукла', 'check1'),
(
 'id2', '03.01.2020', 'хлеб', 'check2'),
(
 'id2', '03.01.2020', 'масло', 'check2'),
(
 'id2', '03.01.2020', 'огурец', 'check2'),
(
 'id1', '01.06.2020', 'машинка', 'check15'),
(
 'id1', '01.06.2020', 'кукла', 'check15'),
(
 'id2', '03.05.2020', 'хлеб', 'check4'),
(
 'id2', '03.05.2020', 'масло', 'check4'),
(
 'id3', '03.05.2020', 'сигареты', 'check65'),
(
 'id3', '03.05.2020', 'спички', 'check65'); 

 -- уникальные пары 
 with t as ( select * from @sales )
   select t1.articul, t2.articul, check_num = min(t1.check_num)
     from t as t1 inner join t as t2 on t1.client_id = t2.client_id and t1.check_num < t2.check_num and t1.articul < t2.articul
     group by t1.articul, t2.articul
;

 -- пары + кол-во таких пар
 with t as ( select * from @sales )
    , t1 as ( select articul1 = t1.articul, articul2 = t2.articul, check_num = min(t1.check_num)
                 from t as t1 inner join t as t2 on t1.client_id = t2.client_id and t1.check_num < t2.check_num and t1.articul < t2.articul
                 group by t1.articul, t2.articul
            )
   select articul1, articul2, count(*) as cnt
     from t1 
          inner join t as t2 on t2.check_num >= t1.check_num and t2.articul = t1.articul1
          inner join t as t3 on t2.client_id = t3.client_id and t3.check_num = t2.check_num and t3.articul = t1.articul2
     group by t1.articul1, t1.articul2
     --having count(*) > 1
     order by cnt desc
;
-- пары с клиентами + кол-во таких у клиента
 with t as ( select * from @sales )
    , t1 as ( select articul1 = t1.articul, articul2 = t2.articul, check_num = min(t1.check_num)
                 from t as t1 inner join t as t2 on t1.client_id = t2.client_id and t1.check_num < t2.check_num and t1.articul < t2.articul
                 group by t1.articul, t2.articul
            )
   select articul1, articul2, count(*) as cnt, t2.client_id
     from t1 
          inner join t as t2 on t2.check_num >= t1.check_num and t2.articul = t1.articul1
          inner join t as t3 on t2.client_id = t3.client_id and t3.check_num = t2.check_num and t3.articul = t1.articul2
     group by t1.articul1, t1.articul2, t2.client_id
     --having count(*) > 1
     order by t1.articul1, t1.articul2, cnt desc, t2.client_id

 -- пары 
 with t as ( select * from @sales )
   select t1.articul, t2.articul, ( sqrt(8*count(*) + 1 ) + 1 ) /2 as cnt
     from t as t1 inner join t as t2 on t1.client_id = t2.client_id and t1.check_num < t2.check_num and t1.articul < t2.articul
     group by t1.articul, t2.articul
 ----    having count(*) > 1
     order by cnt desc
;
-- клиенты c парами
 with t as ( select * from @sales )
   select t1.articul, t2.articul, ( sqrt(8*count(*) + 1 ) + 1 ) /2 as cnt, t1.client_id
     from t as t1 inner join t as t2 on t1.client_id = t2.client_id and t1.check_num < t2.check_num and t1.articul < t2.articul
     group by t1.articul, t2.articul, t1.client_id
 --    having count(*) > 1
     order by t1.articul, t2.articul, cnt desc, client_id