Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 4 / c

Solution

Let

μ = C_{p} (λ)

, put

m = ∣ μ ∣

, and let the first-level

p

-quotient partitions have sizes

n_{0}, \dots, n_{p - 1}

. Then

n = m + p \sum_{j} n_{j} .

d_{p} (n) - d_{p} (m) \leq d_{p} (\sum_{j} n_{j}) \leq \sum_{j} d_{p} (n_{j}) .

For any partition

ν

s

, iterating the core-quotient relation

s = ∣ C_{p} (ν) ∣ + p ∣ Q_{p} (ν) ∣

and the same digit-sum inequality gives

d_{p} (s) \leq \sum_{r \geq 0} ∣ T C_{p} (ν)_{r} ∣.

Apply this to every first-level quotient partition. Their core towers concatenate to levels

r \geq 1

T C_{p} (λ)

, so

d_{p} (n) - d_{p} (m) \leq \sum_{r \geq 1} ∣ T C_{p} (λ)_{r} ∣.

Part a applied to

λ

and to its

p

-core

μ

, whose higher core-tower levels are empty, now gives

(p - 1) (v_{p} (χ^{λ} (1)) - v_{p} (χ^{μ} (1))) = r \geq 1 \sum ∣ T C_{p} (λ)_{r} ∣ - d_{p} (n) + d_{p} (m) \geq 0.

v_{p} (χ^{λ} (1)) \geq v_{p} (χ^{C_{p} (λ)} (1)) .

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