This is the story for the ordinary generating functions for the number of ways of breaking up r identical Russian Dolls into k smaller Russian Dolls, according to the number of atomic dolls, for k small. Note, these were empirically guessed, but it can all be proved rigorously, but frankly, we don't care! The first few generating functions for breaking, 1, identical Russian Dolls into k smaller Russian Dolls, for k from 1 to, 15 using the variable, z, happen to be 2 3 z z z [- -----, -----------------, - ---------------------------, z - 1 (2 z - 1) (z - 1) (z - 1) (3 z - 1) (2 z - 1) 4 z -------------------------------------, (z - 1) (3 z - 1) (2 z - 1) (4 z - 1) 5 z - -----------------------------------------------, (z - 1) (4 z - 1) (3 z - 1) (2 z - 1) (5 z - 1) 6 z ---------------------------------------------------------, (z - 1) (6 z - 1) (4 z - 1) (3 z - 1) (2 z - 1) (5 z - 1) 7 z 8 - -------------------------------------------------------------------, z /( (z - 1) (6 z - 1) (4 z - 1) (3 z - 1) (2 z - 1) (7 z - 1) (5 z - 1) (z - 1) (6 z - 1) (4 z - 1) (3 z - 1) (2 z - 1) (8 z - 1) (7 z - 1) 9 (5 z - 1)), - z /((z - 1) (6 z - 1) (4 z - 1) (3 z - 1) (9 z - 1) (2 z - 1) 10 (8 z - 1) (7 z - 1) (5 z - 1)), z /((z - 1) (6 z - 1) (4 z - 1) (3 z - 1) 11 (9 z - 1) (2 z - 1) (8 z - 1) (7 z - 1) (5 z - 1) (10 z - 1)), - z /( (z - 1) (9 z - 1) (6 z - 1) (7 z - 1) (3 z - 1) (5 z - 1) (2 z - 1) 12 (10 z - 1) (4 z - 1) (11 z - 1) (8 z - 1)), z /((z - 1) (9 z - 1) (6 z - 1) (7 z - 1) (3 z - 1) (5 z - 1) (2 z - 1) (12 z - 1) (10 z - 1) 13 (4 z - 1) (11 z - 1) (8 z - 1)), - z /((z - 1) (9 z - 1) (6 z - 1) (13 z - 1) (7 z - 1) (3 z - 1) (5 z - 1) (2 z - 1) (12 z - 1) (10 z - 1) 14 (4 z - 1) (11 z - 1) (8 z - 1)), z /((z - 1) (9 z - 1) (6 z - 1) (13 z - 1) (7 z - 1) (3 z - 1) (5 z - 1) (2 z - 1) (12 z - 1) (14 z - 1) 15 (10 z - 1) (4 z - 1) (11 z - 1) (8 z - 1)), - z /((z - 1) (9 z - 1) (6 z - 1) (13 z - 1) (7 z - 1) (3 z - 1) (5 z - 1) (15 z - 1) (2 z - 1) (12 z - 1) (14 z - 1) (10 z - 1) (4 z - 1) (11 z - 1) (8 z - 1))] The first few generating functions for breaking, 2, identical Russian Dolls into k smaller Russian Dolls, for k from 1 to, 9 using the variable, z, happen to be 2 2 2 z z z (1 - 5 z + 9 z ) [- -----, -----------------, -------------------------------------, z - 1 (3 z - 1) (z - 1) (z - 1) (6 z - 1) (3 z - 1) (2 z - 1) 3 2 z (3 - 22 z + 64 z ) - ------------------------------------------------, (z - 1) (6 z - 1) (4 z - 1) (2 z - 1) (10 z - 1) 3 2 3 4 z (1 - 17 z + 146 z - 452 z + 700 z ) 4 - ---------------------------------------------------------------------, z (z - 1) (7 z - 1) (3 z - 1) (15 z - 1) (2 z - 1) (10 z - 1) (4 z - 1) 2 3 4 5 (-6 + 191 z - 2758 z + 19272 z - 69444 z + 114345 z )/((z - 1) (6 z - 1) (7 z - 1) (3 z - 1) (5 z - 1) (15 z - 1) (2 z - 1) (21 z - 1) (11 z - 1)), 4 2 3 4 5 6 z (1 - 50 z + 1215 z - 16585 z + 145584 z - 853843 z + 2872262 z 7 8 - 4587104 z + 4139520 z )/((z - 1) (10 z - 1) (6 z - 1) (5 z - 1) (11 z - 1) (3 z - 1) (8 z - 1) (2 z - 1) (21 z - 1) (16 z - 1) (4 z - 1) 5 2 3 4 (28 z - 1)), - 5 z (-2 + 180 z - 7591 z + 188079 z - 3000892 z 5 6 7 8 + 31965124 z - 225147136 z + 1009743808 z - 2697596928 z 9 + 3451650048 z )/((z - 1) (16 z - 1) (12 z - 1) (8 z - 1) (6 z - 1) (4 z - 1) (22 z - 1) (3 z - 1) (36 z - 1) (15 z - 1) (2 z - 1) (7 z - 1) 5 2 3 4 (10 z - 1) (28 z - 1)), z (1 - 115 z + 5920 z - 163563 z + 2177270 z 5 6 7 8 + 6505653 z - 859897070 z + 17925925819 z - 213914575475 z 9 10 11 + 1685112555030 z - 9242910730926 z + 35229069992376 z 12 13 - 83118203178120 z + 85478119927200 z )/((z - 1) (12 z - 1) (6 z - 1) (21 z - 1) (4 z - 1) (29 z - 1) (22 z - 1) (3 z - 1) (36 z - 1) (5 z - 1) (9 z - 1) (15 z - 1) (45 z - 1) (2 z - 1) (7 z - 1) (11 z - 1) (10 z - 1) (17 z - 1))] The first few generating functions for breaking, 3, identical Russian Dolls into k smaller Russian Dolls, for k from 1 to, 8 using the variable, z, happen to be 2 2 z z (3 z - 1) z (6 z - 1) [- -----, ---------------------------, ------------------------------, z - 1 (z - 1) (2 z - 1) (4 z - 1) (4 z - 1) (2 z - 1) (10 z - 1) 2 2 3 4 z (1 - 32 z + 356 z - 1528 z + 2400 z ) 3 -----------------------------------------------------------, - z (-7 (z - 1) (4 z - 1) (20 z - 1) (2 z - 1) (8 z - 1) (10 z - 1) 2 3 4 5 6 + 451 z - 11843 z + 155991 z - 1135566 z + 4643810 z - 10023900 z 7 + 9030000 z )/((z - 1) (15 z - 1) (35 z - 1) (5 z - 1) (3 z - 1) (8 z - 1) 3 2 (2 z - 1) (20 z - 1) (4 z - 1) (7 z - 1)), - z (3 - 377 z + 20097 z 3 4 5 6 7 - 574005 z + 9876180 z - 107315313 z + 743969037 z - 3245301427 z 8 9 10 + 8438288235 z - 11303359830 z + 5032427400 z )/((z - 1) (12 z - 1) (6 z - 1) (4 z - 1) (3 z - 1) (5 z - 1) (56 z - 1) (15 z - 1) (2 z - 1) 3 2 (7 z - 1) (11 z - 1) (26 z - 1) (35 z - 1)), - z (-1 + 231 z - 23340 z 3 4 5 6 + 1334461 z - 48764647 z + 1208038521 z - 20968112062 z 7 8 9 + 260417241263 z - 2330180428878 z + 14866342335420 z 10 11 12 - 65827330339752 z + 192842402616864 z - 337643002730880 z 13 + 265915582732800 z )/((z - 1) (12 z - 1) (9 z - 1) (84 z - 1) (6 z - 1) (11 z - 1) (4 z - 1) (42 z - 1) (3 z - 1) (20 z - 1) (2 z - 1) (26 z - 1) 4 2 (21 z - 1) (56 z - 1) (5 z - 1) (10 z - 1)), z (25 - 8698 z + 1338980 z 3 4 5 6 - 118032064 z + 6666190375 z - 254976585078 z + 6803914366548 z 7 8 9 - 128884869410168 z + 1744699224991712 z - 16803385919266304 z 10 11 + 113583226020721152 z - 525113780330354688 z 12 13 + 1578842249448849408 z - 2780458546102272000 z 14 + 2186871770500300800 z )/((z - 1) (16 z - 1) (8 z - 1) (6 z - 1) (21 z - 1) (4 z - 1) (84 z - 1) (3 z - 1) (36 z - 1) (5 z - 1) (9 z - 1) (2 z - 1) (32 z - 1) (42 z - 1) (10 z - 1) (64 z - 1) (120 z - 1) (20 z - 1))] this took, 163.230, seconds