• Mr. Mukesh Yadav

Here we try to study fundamental additive decomposition process i.e. the representation of positive integers by sums of other positive integers. If we consider order of term into account the sum is called composition otherwise consider it partition. Notation used here, C (i) is no. of compositions of i and P (i) for the no of partitions. Thus, compositions (number three) are 3,1 + 2,2 + 1,1 + 1 + 1 which implies C (3) = 4 and P(3) = 3 since 3,2 + 1,1 + 1 + 1 are partition of no. three here. Compositions If we consider order of term into account the sum is called compositions In Particular,Take number four then no of partitions are : (4),(13),(2^2),(1^2 2),(1^4) ; there are eight compositions of 4 are: (4),(13),(31),(22),(112),(121),(211),(1111). Definition . We let C(m,n) = the no. of compositions of n into m parts (exactly) Definition . We let C(N,M,n) total no.of of compositions of n with M parts exactly, each ≤ N. Obviously C (N,M,n) = C (M,n) whenever N≥ n. Interestingly enough, C (N,M,n) possesses symmetry properties . Definition A partition of (α_1,α_2,…,α_r) is a set of vectors (β_1^((i)),… ,β_r^((i))) , 1 ≤i≤s (order disregarded), such that ∑_(i=1)^s▒( β_1^((i)),… ,β_r^((j)))=(α_1,α_2,… ,α_r) (here as explained earlier all vectors have non-negative integral coordinates not all zero). If we consider order of term into account , we call (β_1^((1)),…,β_r^((1))) , . . . , (β_1^((s〉),… ,β_r^((s))) a composition of (α_1… ,α_s). Definition We let P = (α_1,α_2,…,α_r:m) be partition of (α_1,α_2,…,α_r) with m parts, and C (α_1,α_2,… ,α_r;m) be the no. of compositions of (α_1,α_2….,α_r) with m parts. Thus P = (2, 1, 1; 2)= 5, since there are five partitions of (2, 1, 1) into two parts: (2,1,0)(0,0,1) , (2.0. 1)(0, 1,0) , (2,0,0)(0, 1,1) , (1,1,0)(1,0,1) , (1, 1, 1)(1,0 0) ; C (2,1,1; 2)= 10 since each of the rive partitions produces two compositions. Definition We let P(α_1,…,α_r) (respectively C (α_1, … . α_r)) is the total no. of partitions (respectively compositions) of (α_1,α_2,… ,α_r) .For later convenience we define C (0,0,…,0) = 1/2. Observe that ∑_(m≥1)^ ▒P (α_1,…,α_r:m)=P(α_1,…,α_r ), and ∑_(m≥1)^ ▒C(α_1,…,α_r;m)=C (α_1,…,α_r) .

Compositions,Co-efficient, Partition, identities , decomposition.

"PARTITIONS IN PARTITION THEORY", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.9, Issue 2, page no.b66-b70, February-2022, Available :http://www.jetir.org/papers/JETIR2202109.pdf

"PARTITIONS IN PARTITION THEORY", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.9, Issue 2, page no. ppb66-b70, February-2022, Available at : http://www.jetir.org/papers/JETIR2202109.pdf