Infinitely Many Ramanujan-type Congruences for p4,1(n) mod 11 - Sayan Kumar Mitra Class 11 Rajendra Vidyalaya Jamshedpur

This paper presents a proof of a new infinite family of Ramanujan-type congruences for the partition function p4,1(n), which counts partitions where odd parts have 4 colors and even parts have 1 color. I proved: p4,1(11n+10) ≡ 0 (mod 11) for all n ≥ 0. The generating function is G4,1(q) = ∏k odd (1-q^k)⁻⁴ ∏k even (1-q^k)⁻¹. Using Fermat's Little Theorem (1-q^k)¹¹ ≡ 1-q¹¹ᵏ (mod 11), we reduce G4,1(q) to an eta-quotient form: A(q) = ∏odd (1-q^k)^7 ∏ₑᵥₑₙ (1-q^k)¹⁰ = q-13/24 η(q)^7 η(q^2)^3 Let f(q)=η(q)^7 η(q^2)^3, a modular form of weight 5 on Γ0(8). By applying the U₁₁ operator and twisting, the congruence reduces to showing the first B=60 Fourier coefficients vanish modulo 11, where B = K/12 * [SL2:Γ0(88)] = 60 is the Sturm bound for level 88. Computational verification was performed in Python up to n=1000, with examples: p4,1(10)=2893 ≡ 0, p4,1(21)=443256 ≡ 0 (mod 11). This extends my previous result p4,1(7n+4) ≡ 0 (mod 7). Author: Sayan Kumar Mitra, Class 11, Rajendra Vidyalaya, Jamshedpur, India. This work was done independently.