Combinatorial Magic Right-angle Triangle Characterization on Partial 𝒕 △ − Symmetric Contraction Semigroups

Let Mn = {m1 , m2 , … mn } be n-element distinct non-negative integer, let Pn, Tn, In, CIn, ECIn, MICn be partial transformation semigroup, full transformation semigroup, symmetric inverse semigroup, contraction (one-one) symmetric inverse, contraction idempotent, magic right-angle triangle contraction symmetric inverse respectively. The semigroup (Sn,∗) of any given partial contraction one-one transformation α ∈ Sn: D(α) ⊆ Mn ⟶ I(α) is said to be t △-symmetric if D(α) ⊆ Mn: cn ⊆ bn where |cn | ≤ |bn | such that (Sn,∗) is closed under basic counting principle (sums), contains a constant (identity) element and generate magic right-angle triangles using some combinatorial parameters. This paper investigates some combinatorial parameters (r(α), b(α), k +(α), and k −(α)) to characterize magic right-angle triangle for all m, n ∈ Mn, |αm − αn| ≤ |m −n| is contraction mapping such that αm, αn ∈ D(α), provided that any element in D(α) is not assumed to be mapped to empty ∅ as contraction. For a given α ∈ Sn there exist t ∈ S:{t = |Max(n, w+)| ∗ |Min(n, w−)|} for all n ≥ 1; n ∈ N then (n; k +(α), k −(α)) = ∑ ( 2 k−1 k +n −1 ) k n=1 such that ECIn is t ∆ − symmetric, also if |E(S)| = n(n+1) 2 + 1, then f(n; p, m) = ∑ ( n m ) n(n−1) m n m=2 for all n ≥ 2; n, m ∈ Mn.