Some Special Integer Partitions Generated by a Family of Functions

MATTE, M. L.

doi:10.5540/tcam.2023.024.04.00717

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Articles • Trends Comput. Appl. Math. 24 (4) • Oct-Dec 2023 • https://doi.org/10.5540/tcam.2023.024.04.00717 copy

Some Special Integer Partitions Generated by a Family of Functions

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

In this work, inspired by Ramanujan’s fifth order Mock Theta function f ₁( q ), we define a collection of functions and look at them as generating functions for partitions of some integer n containing at least m parts equal to each one of the numbers from 1 to its greatest part s , with no gaps. We set a two-line matrix representation for these partitions for any m ≥ 2 and collect the values of the sum of the entries in the second line of those matrices. These sums contain information about some parts of the partitions, which lead us to closed formulas for the number of partitions generated by our functions, and partition identities involving other simpler and well known partition functions.

Keywords:
integer partition; mock theta function; matrix representation; partition identity

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45

0	1
1	0	0
2	0	0	0
3	0	0	0	0
4	0	0	0	0	1
5	0	0	0	0	1	0
6	0	0	0	0	1	0	0
7	0	0	0	0	1	0	0	0
8	0	0	0	0	1	0	0	0	0
9	0	0	0	0	1	0	0	0	0	0
10	0	0	0	0	1	0	0	0	0	0	0
11	0	0	0	0	1	0	0	0	0	0	0	0
12	0	0	0	0	1	0	0	0	0	0	0	0	1
13	0	0	0	0	1	0	0	0	0	0	0	0	1	0
14	0	0	0	0	1	0	0	0	0	0	0	0	1	1	0
15	0	0	0	0	1	0	0	0	0	0	0	0	1	1	0	0
16	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	0	0
17	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	0	0	0
18	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	0	0	0
19	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0
20	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0
21	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0
22	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0
23	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
24	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	1
25	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	1	0
26	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	1	1	0
27	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	1	1	1	0
28	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	0	0
29	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	1	0	0
30	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	1	0	0
31	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	2	0	0	0
32	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	1	0	0	0
33	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	2	1	0	0	0
34	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	2	0	0	0	0
35	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	3	1	0	0	0	0
36	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	2	1	0	0	0	0
37	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	3	2	0	0	0	0	0
38	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	4	4	3	1	0	0	0	0	0
39	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	4	4	4	2	1	0	0	0	0	0
40	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	3	2	0	0	0	0	0	1
41	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	4	3	1	0	0	0	0	1	0
42	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	4	2	1	0	0	0	1	1	0
43	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	5	3	2	0	0	0	1	1	1	0
44	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	6	4	3	1	0	0	1	1	2	1	0
45	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	6	5	4	2	1	0	1	1	2	2	0	0

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45

0	1
1	0	0
2	0	0	0
3	0	0	0	1
4	0	0	0	0	0
5	0	0	0	0	0	1
6	0	0	0	0	0	1	0
7	0	0	0	0	0	1	0	0
8	0	0	0	0	0	1	0	0	0
9	0	0	0	0	0	1	0	0	0	0
10	0	0	0	0	0	1	0	0	0	0	0
11	0	0	0	0	0	1	0	0	0	0	0	0
12	0	0	0	0	0	1	0	0	0	0	0	0	0
13	0	0	0	0	0	1	0	0	0	0	0	0	0	0
14	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
15	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1
16	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	0
17	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	0
18	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	0	0
19	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0	0
20	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0	0	0
21	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0
22	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0
23	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0
24	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0
25	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0
26	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
27	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0
28	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0	0
29	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0
30	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	1
31	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	1	0
32	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	1	1	0
33	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	1	1	1	0
34	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	1	1	2	0	0
35	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	1	1	2	1	0	0
36	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	1	1	2	2	1	0	0
37	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	2	2	0	0	0
38	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	2	3	1	0	0	0
39	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	3	2	1	0	0	0
40	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	3	3	2	0	0	0	0
41	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	3	3	1	0	0	0	0
42	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	3	4	2	1	0	0	0	0
43	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	4	3	2	0	0	0	0	0
44	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	4	4	3	1	0	0	0	0	0
45	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	4	4	2	1	0	0	0	0	0

Sociedade Brasileira de Matemática Aplicada e Computacional - SBMAC Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163, Cep: 13561-120 - SP / São Carlos - Brasil, +55 (16) 3412-9752 - São Carlos - SP - Brazil
E-mail: sbmac@sbmac.org.br

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$f_{*}^{m} (q)$	$p_{[s]}^{m [s]} (n, 2)$
$f_{*}^{2} (q)$	(0 , 0 , 0 , 1 , 0 , 0 , 0 , 1 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 5 , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 , 6 , 5 , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 , 7 , 6 , 6 , 5 , 5 , . . . )
$f_{*}^{3} (q)$	(0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , . . . )
$f_{*}^{4} (q)$	(0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , . . . )

$P_{[s]}^{4 [s]} (189, 2)$	$P_{[s]}^{4 [s]} (191, 2)$
(9 , 9 , 9 , 9 , 8 , 8 , 8 , 8 , . . . , 1 , 1 , 1 , 1 , 8 , 1 )	(9 , 9 , 9 , 9 , 8 , 8 , 8 , 8 , . . . , 1 , 1 , 1 , 1 , 9 , 2 )
(9 , 9 , 9 , 9 , 8 , 8 , 8 , 8 , . . . , 1 , 1 , 1 , 1 , 7 , 2 )	(9 , 9 , 9 , 9 , 8 , 8 , 8 , 8 , . . . , 1 , 1 , 1 , 1 , 8 , 3 )
(9 , 9 , 9 , 9 , 8 , 8 , 8 , 8 , . . . , 1 , 1 , 1 , 1 , 6 , 3 )	(9 , 9 , 9 , 9 , 8 , 8 , 8 , 8 , . . . , 1 , 1 , 1 , 1 , 7 , 4 )
(9 , 9 , 9 , 9 , 8 , 8 , 8 , 8 , . . . , 1 , 1 , 1 , 1 , 5 , 4 )	(9 , 9 , 9 , 9 , 8 , 8 , 8 , 8 , . . . , 1 , 1 , 1 , 1 , 6 , 5 )

91 ± i	$P_{[s]}^{4 [s]} (91 \pm i, 2)$	$p_{[s]}^{4 [s]} (91 \pm i, 2)$
86	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 1 , 1 )	1
87	(6, 6, 6, 6, 5, 5, 5, 5, . . . ,1, 1, 1, 1, 2 , 1 )	1
88	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 3 , 1 )	2
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 2 , 2 )
89	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 4 , 1 )	2
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 3 , 2 )
90	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 5 , 1 )	3
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 4 , 2 )
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 3 , 3 )
91	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 6 , 1 )	3
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 5 , 2 )
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 4 , 3 )
92	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 6 , 2 )	3
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 5 , 3 )
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 4 , 4 )
93	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 6 , 3 )	2
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 5 , 4 )
94	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 6 , 4 )	2
	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 5 , 5 )
95	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 6 , 5 )	1
96	(6, 6, 6, 6, 5, 5, 5, 5, . . . , 1, 1, 1, 1, 6 , 6 )	1

$f_{*}^{m} (q)$	$p_{[s]}^{m [s]} (n, 3)$
$f_{*}^{3} (q)$	(0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 6 , 6 , 6 , 6 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 7 , 8 , 8 , 8 , 7 , 7 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 , 10 , 10 , 10 , 9 , 8 , 7 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 8 , 10 , 11 , 12 , 12 , 13 , 12 , 12 , 11 , 10 , 8 , 7 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 8 , 10 , 12 , 13 , 14 , 15 , 15 , 15 , 15 , 14 , 13 , 12 , 10 , 8 , 7 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , . . . )
$f_{*}^{4} (q)$	(0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 6 , 6 , 6 , 6 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 7 , 8 , 8 , 8 , 7 , 7 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 , 10 , 10 , 10 , 9 , 8 , 7 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 8 , 10 , 11 , 12 , 12 , 13 , 12 , 12 , 11 , 10 , 8 , . . . )
$f_{*}^{5} (q)$	(0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 3 , 3 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 6 , 6 , 6 , 6 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 7 , 8 , 8 , 8 , 7 , 7 , 5 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 7 , 8 , 9 , 10 , 10 , 10 , 10 , 9 , 8 , 7 , 5 , 4 , . . . )

$P_{[s]}^{3 [s]} (52, 3)$	$P_{[s]}^{3 [s]} (56, 3)$
(5 , 5 , 5 , 4 , 4 , 4 , 3 , 3 , 3 , 2 , 2 , 2 , 1 , 1 , 1 , 5 , 1 , 1 )	(5 , 5 , 5 , 4 , 4 , 4 , 3 , 3 , 3 , 2 , 2 , 2 , 1 , 1 , 1 , 1 , 5 , 5 )
(5 , 5 , 5 , 4 , 4 , 4 , 3 , 3 , 3 , 2 , 2 , 2 , 1 , 1 , 1 , 4 , 2 , 1 )	(5 , 5 , 5 , 4 , 4 , 4 , 3 , 3 , 3 , 2 , 2 , 2 , 1 , 1 , 1 , 2 , 4 , 5 )
(5 , 5 , 5 , 4 , 4 , 4 , 3 , 3 , 3 , 2 , 2 , 2 , 1 , 1 , 1 , 3 , 3 , 1 )	(5 , 5 , 5 , 4 , 4 , 4 , 3 , 3 , 3 , 2 , 2 , 2 , 1 , 1 , 1 , 3 , 3 , 5 )
(5 , 5 , 5 , 4 , 4 , 4 , 3 , 3 , 3 , 2 , 2 , 2 , 1 , 1 , 1 , 3 , 2 , 2 )	(5 , 5 , 5 , 4 , 4 , 4 , 3 , 3 , 3 , 2 , 2 , 2 , 1 , 1 , 1 , 3 , 4 , 4 )

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45

0	1
1	0	0
2	0	0	0
3	0	0	0	0
4	0	0	0	0	1
5	0	0	0	0	1	0
6	0	0	0	0	1	0	0
7	0	0	0	0	1	0	0	0
8	0	0	0	0	1	0	0	0	0
9	0	0	0	0	1	0	0	0	0	0
10	0	0	0	0	1	0	0	0	0	0	0
11	0	0	0	0	1	0	0	0	0	0	0	0
12	0	0	0	0	1	0	0	0	0	0	0	0	1
13	0	0	0	0	1	0	0	0	0	0	0	0	1	0
14	0	0	0	0	1	0	0	0	0	0	0	0	1	1	0
15	0	0	0	0	1	0	0	0	0	0	0	0	1	1	0	0
16	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	0	0
17	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	0	0	0
18	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	0	0	0
19	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0
20	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0
21	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0
22	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0
23	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
24	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	1
25	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	1	0
26	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	1	1	0
27	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	1	1	1	0
28	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	0	0
29	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	1	0	0
30	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	1	0	0
31	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	2	0	0	0
32	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	1	0	0	0
33	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	2	1	0	0	0
34	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	2	0	0	0	0
35	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	3	1	0	0	0	0
36	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	2	1	0	0	0	0
37	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	3	2	0	0	0	0	0
38	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	4	4	3	1	0	0	0	0	0
39	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	4	4	4	2	1	0	0	0	0	0
40	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	3	2	0	0	0	0	0	1
41	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	4	3	1	0	0	0	0	1	0
42	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	4	2	1	0	0	0	1	1	0
43	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	5	3	2	0	0	0	1	1	1	0
44	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	6	4	3	1	0	0	1	1	2	1	0
45	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	6	5	4	2	1	0	1	1	2	2	0	0

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45

0	1
1	0	0
2	0	0	0
3	0	0	0	1
4	0	0	0	0	0
5	0	0	0	0	0	1
6	0	0	0	0	0	1	0
7	0	0	0	0	0	1	0	0
8	0	0	0	0	0	1	0	0	0
9	0	0	0	0	0	1	0	0	0	0
10	0	0	0	0	0	1	0	0	0	0	0
11	0	0	0	0	0	1	0	0	0	0	0	0
12	0	0	0	0	0	1	0	0	0	0	0	0	0
13	0	0	0	0	0	1	0	0	0	0	0	0	0	0
14	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
15	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1
16	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	0
17	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	0
18	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	0	0
19	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0	0
20	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0	0	0
21	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0
22	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0
23	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0
24	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0
25	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0
26	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
27	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0
28	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0	0
29	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0
30	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	1
31	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	1	0
32	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	1	1	0
33	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	1	1	1	0
34	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	1	1	2	0	0
35	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	1	1	2	1	0	0
36	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	1	1	2	2	1	0	0
37	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	2	2	0	0	0
38	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	2	3	1	0	0	0
39	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	3	2	1	0	0	0
40	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	3	3	2	0	0	0	0
41	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	3	3	1	0	0	0	0
42	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	3	4	2	1	0	0	0	0
43	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	4	3	2	0	0	0	0	0
44	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	4	4	3	1	0	0	0	0	0
45	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	4	4	2	1	0	0	0	0	0

$P_{[s]}^{3 [s]} (52, 3)$	$P_{[s]}^{3 [s]} (142, 3)$	P (4 , at most 3 parts)
(5 , 5 , 5 , · · · , 2 , 2 , 2 , 1 , 1 , 1 , 5 , 1 , 1 )	(9 , 9 , 9 , · · · , 1 , 1 , 1 , 5 , 1 , 1 )	(4)
(5 , 5 , 5 , · · · , 2 , 2 , 2 , 1 , 1 , 1 , 4 , 2 , 1 )	(9 , 9 , 9 , · · · , 1 , 1 , 1 , 4 , 2 , 1 )	(3 , 1)
(5 , 5 , 5 , · · · , 2 , 2 , 2 , 1 , 1 , 1 , 3 , 3 , 1 )	(9 , 9 , 9 , · · · , 1 , 1 , 1 , 3 , 3 , 1 )	(2 , 2)
(5 , 5 , 5 , · · · , 2 , 2 , 2 , 1 , 1 , 1 , 3 , 2 , 2 )	(9 , 9 , 9 , · · · , 1 , 1 , 1 , 3 , 2 , 2 )	(2 , 1 , 1)

$f_{*}^{m} (q)$	$p_{[s]}^{m [s]} (n, 4)$
$f_{*}^{4} (q)$	(0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 5 , 7 , 7 , 8 , 7 , 7 , 5 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 8 , 9 , 11 , 11 , 12 , 11 , 11 , 9 , 8 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 9 , 10 , 13 , 14 , 16 , 16 , 18 , 16 , 16 , 14 , 13 , 10 , 9 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 9 , 11 , 14 , 16 , 19 , 20 , 23 , 23 , 24 , 23 , 23 , 20 , 19 , 16 , 14 , 11 , 9 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 9 , 11 , 15 , 17 , 21 , 23 , 27 , 28 , 31 , 31 , 33 , 31 , 31 , 28 , 27 , 23 , 21 , 17 , 15 , 11 , 9 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , . . . )
$f_{*}^{5} (q)$	(0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 5 , 7 , 7 , 8 , 7 , 7 , 5 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 8 , 9 , 11 , 11 , 12 , 11 , 11 , 9 , 8 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 9 , 10 , 13 , 14 , 16 , 16 , 18 , 16 , 16 , 14 , 13 , 10 , 9 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 9 , 11 , 14 , 16 , 19 , 20 , 23 , 23 , 24 , 23 , 23 , 20 , 19 , 16 , 14 , 11 , 9 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , . . . )
$f_{*}^{6} (q)$	(0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 3 , 2 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 4 , 4 , 5 , 4 , 4 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 5 , 7 , 7 , 8 , 7 , 7 , 5 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 8 , 9 , 11 , 11 , 12 , 11 , 11 , 9 , 8 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 2 , 3 , 5 , 6 , 9 , 10 , 13 , 14 , 16 , 16 , 18 , 16 , 16 , 14 , 13 , 10 , 9 , 6 , 5 , 3 , 2 , 1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , . . . )

$P_{[s]}^{4 [s]} (59, 4)$	$P_{[s]}^{4 [s]} (61, 4)$
(4 , 4 , 4 , 4 , 4 , 3 , . . . , 1 , 1 , 1 , 1 , 1 , 4 , 2 , 2 , 1 )	(4 , 4 , 4 , 4 , 4 , 3 , . . . , 1 , 1 , 1 , 1 , 1 , 4 , 3 , 3 , 1 )
(4 , 4 , 4 , 4 , 4 , 3 , . . . , 1 , 1 , 1 , 1 , 1 , 4 , 3 , 1 , 1 )	(4 , 4 , 4 , 4 , 4 , 3 , . . . , 1 , 1 , 1 , 1 , 1 , 4 , 4 , 2 , 1 )
(4 , 4 , 4 , 4 , 4 , 3 , . . . , 1 , 1 , 1 , 1 , 1 , 3 , 3 , 2 , 1 )	(4 , 4 , 4 , 4 , 4 , 3 , . . . , 1 , 1 , 1 , 1 , 1 , 4 , 3 , 2 , 2 )
(4 , 4 , 4 , 4 , 4 , 3 , . . . , 1 , 1 , 1 , 1 , 1 , 3 , 2 , 2 , 2 )	(4 , 4 , 4 , 4 , 4 , 3 , . . . , 1 , 1 , 1 , 1 , 1 , 3 , 3 , 3 , 2 )

$P_{[s]}^{5 [s]} (114, 4)$	$P_{[s]}^{5 [s]} (284, 4)$	P (5 , at most 4 parts)
(6 , 6 , 6 , 6 , 6 , · · · , 1 , 1 , 1 , 1 , 1 , 6 , 1 , 1 , 1 )	(10 , 10 , 10 , 10 , 10 , · · · , 1 , 1 , 1 , 1 , 1 , 6 , 1 , 1 , 1 )	(5)
(6 , 6 , 6 , 6 , 6 , · · · , 1 , 1 , 1 , 1 , 1 , 5 , 2 , 1 , 1 )	(10 , 10 , 10 , 10 , 10 , · · · , 1 , 1 , 1 , 1 , 1 , 5 , 2 , 1 , 1 )	(4 , 1)
(6 , 6 , 6 , 6 , 6 , · · · , 1 , 1 , 1 , 1 , 1 , 4 , 3 , 1 , 1 )	(10 , 10 , 10 , 10 , 10 , · · · , 1 , 1 , 1 , 1 , 1 , 4 , 3 , 1 , 1 )	(3 , 2)
(6 , 6 , 6 , 6 , 6 , · · · , 1 , 1 , 1 , 1 , 1 , 4 , 2 , 2 , 1 )	(10 , 10 , 10 , 10 , 10 , · · · , 1 , 1 , 1 , 1 , 1 , 4 , 2 , 2 , 1 )	(3 , 1 , 1)
(6 , 6 , 6 , 6 , 6 , · · · , 1 , 1 , 1 , 1 , 1 , 3 , 3 , 2 , 1 )	(10 , 10 , 10 , 10 , 10 , · · · , 1 , 1 , 1 , 1 , 1 , 3 , 3 , 2 , 1 )	(2 , 2 , 1)
(6 , 6 , 6 , 6 , 6 , · · · , 1 , 1 , 1 , 1 , 1 , 3 , 2 , 2 , 2 )	(10 , 10 , 10 , 10 , 10 , · · · , 1 , 1 , 1 , 1 , 1 , 3 , 2 , 2 , 2 )	(2 , 1 , 1 , 1)

n = 1	$P_{[s]}^{4 [s]} (8, 4)$	P (10 , exactly 4 distinct parts in (4))
	(1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 )	(4 , 3 , 2 , 1)
n = 2	$P_{[s]}^{4 [s]} (18, 4)$	P (12 , exactly 4 distinct parts in (5))
	(2 , 2 , 2 , 2 , 1 , 1 , 1 , 1 , 2 , 2 , 1 , 1 )	(5 , 4 , 2 , 1)
n = 3	$P_{[s]}^{4 [s]} (32, 4)$	P (14 , exactly 4 distinct parts in (6))
	(3 , 3 , 3 , 3 , 2 , 2 , 2 , 2 , 1 , 1 , 1 , 1 , 3 , 3 , 1 , 1 )	(6 , 5 , 2 , 1)
	(3 , 3 , 3 , 3 , 2 , 2 , 2 , 2 , 1 , 1 , 1 , 1 , 3 , 2 , 2 , 1 )	(6 , 4 , 3 , 1)
	(3 , 3 , 3 , 3 , 2 , 2 , 2 , 2 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 2 )	(5 , 4 , 3 , 2)

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45

0	1
1	0	0
2	0	0	0
3	0	0	0	0
4	0	0	0	0	1
5	0	0	0	0	1	0
6	0	0	0	0	1	0	0
7	0	0	0	0	1	0	0	0
8	0	0	0	0	1	0	0	0	0
9	0	0	0	0	1	0	0	0	0	0
10	0	0	0	0	1	0	0	0	0	0	0
11	0	0	0	0	1	0	0	0	0	0	0	0
12	0	0	0	0	1	0	0	0	0	0	0	0	1
13	0	0	0	0	1	0	0	0	0	0	0	0	1	0
14	0	0	0	0	1	0	0	0	0	0	0	0	1	1	0
15	0	0	0	0	1	0	0	0	0	0	0	0	1	1	0	0
16	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	0	0
17	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	0	0	0
18	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	0	0	0
19	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0
20	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0
21	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0
22	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0
23	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
24	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	1
25	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	1	0
26	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	1	1	0
27	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	1	1	1	0
28	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	0	0
29	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	1	0	0
30	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	1	0	0
31	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	2	0	0	0
32	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	1	0	0	0
33	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	2	1	0	0	0
34	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	2	0	0	0	0
35	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	3	1	0	0	0	0
36	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	2	1	0	0	0	0
37	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	3	2	0	0	0	0	0
38	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	4	4	3	1	0	0	0	0	0
39	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	4	4	4	2	1	0	0	0	0	0
40	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	3	2	0	0	0	0	0	1
41	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	4	3	1	0	0	0	0	1	0
42	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	4	2	1	0	0	0	1	1	0
43	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	5	3	2	0	0	0	1	1	1	0
44	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	6	4	3	1	0	0	1	1	2	1	0
45	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	6	5	4	2	1	0	1	1	2	2	0	0

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45

0	1
1	0	0
2	0	0	0
3	0	0	0	1
4	0	0	0	0	0
5	0	0	0	0	0	1
6	0	0	0	0	0	1	0
7	0	0	0	0	0	1	0	0
8	0	0	0	0	0	1	0	0	0
9	0	0	0	0	0	1	0	0	0	0
10	0	0	0	0	0	1	0	0	0	0	0
11	0	0	0	0	0	1	0	0	0	0	0	0
12	0	0	0	0	0	1	0	0	0	0	0	0	0
13	0	0	0	0	0	1	0	0	0	0	0	0	0	0
14	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
15	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1
16	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	0
17	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	0
18	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	0	0
19	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0	0
20	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0	0	0
21	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0
22	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0
23	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0
24	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0
25	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0
26	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
27	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0
28	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0	0
29	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0
30	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	1
31	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	1	0
32	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	1	1	0
33	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	1	1	1	0
34	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	1	1	2	0	0
35	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	1	1	2	1	0	0
36	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	1	1	2	2	1	0	0
37	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	2	2	0	0	0
38	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	2	3	1	0	0	0
39	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	3	2	1	0	0	0
40	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	3	3	2	0	0	0	0
41	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	3	3	1	0	0	0	0
42	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	3	4	2	1	0	0	0	0
43	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	4	3	2	0	0	0	0	0
44	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	4	4	3	1	0	0	0	0	0
45	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	4	4	2	1	0	0	0	0	0

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45

0	1
1	0	0
2	0	0	0
3	0	0	0	0
4	0	0	0	0	1
5	0	0	0	0	1	0
6	0	0	0	0	1	0	0
7	0	0	0	0	1	0	0	0
8	0	0	0	0	1	0	0	0	0
9	0	0	0	0	1	0	0	0	0	0
10	0	0	0	0	1	0	0	0	0	0	0
11	0	0	0	0	1	0	0	0	0	0	0	0
12	0	0	0	0	1	0	0	0	0	0	0	0	1
13	0	0	0	0	1	0	0	0	0	0	0	0	1	0
14	0	0	0	0	1	0	0	0	0	0	0	0	1	1	0
15	0	0	0	0	1	0	0	0	0	0	0	0	1	1	0	0
16	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	0	0
17	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	0	0	0
18	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	0	0	0
19	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0
20	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0
21	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0
22	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0
23	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
24	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	1
25	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	1	0
26	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	1	1	0
27	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	1	1	1	0
28	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	0	0
29	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	1	0	0
30	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	1	0	0
31	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	2	0	0	0
32	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	1	0	0	0
33	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	2	1	0	0	0
34	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	2	0	0	0	0
35	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	3	1	0	0	0	0
36	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	2	1	0	0	0	0
37	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	3	2	0	0	0	0	0
38	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	4	4	3	1	0	0	0	0	0
39	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	4	4	4	2	1	0	0	0	0	0
40	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	3	2	0	0	0	0	0	1
41	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	4	3	1	0	0	0	0	1	0
42	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	4	2	1	0	0	0	1	1	0
43	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	5	3	2	0	0	0	1	1	1	0
44	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	6	4	3	1	0	0	1	1	2	1	0
45	0	0	0	0	1	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	2	3	3	4	4	5	5	6	5	4	2	1	0	1	1	2	2	0	0

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45

0	1
1	0	0
2	0	0	0
3	0	0	0	1
4	0	0	0	0	0
5	0	0	0	0	0	1
6	0	0	0	0	0	1	0
7	0	0	0	0	0	1	0	0
8	0	0	0	0	0	1	0	0	0
9	0	0	0	0	0	1	0	0	0	0
10	0	0	0	0	0	1	0	0	0	0	0
11	0	0	0	0	0	1	0	0	0	0	0	0
12	0	0	0	0	0	1	0	0	0	0	0	0	0
13	0	0	0	0	0	1	0	0	0	0	0	0	0	0
14	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
15	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1
16	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	0
17	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	0
18	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	0	0
19	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0	0
20	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0	0	0
21	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0
22	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	0	0	0
23	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0
24	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	0	0	0	0	0
25	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0
26	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	0	0	0
27	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0
28	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	0	0	0	0	0
29	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0
30	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	1
31	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	1	0
32	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	1	1	0
33	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	1	1	1	0
34	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	1	1	2	0	0
35	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	1	1	2	1	0	0
36	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	0	0	0	0	1	1	2	2	1	0	0
37	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	2	2	0	0	0
38	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	0	0	0	1	1	2	2	3	1	0	0	0
39	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	3	2	1	0	0	0
40	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	2	2	3	3	2	0	0	0	0
41	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	3	3	1	0	0	0	0
42	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	2	2	3	3	4	2	1	0	0	0	0
43	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	4	3	2	0	0	0	0	0
44	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	3	3	4	4	3	1	0	0	0	0	0
45	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	2	2	3	3	4	4	4	2	1	0	0	0	0	0