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The implication is false. ChooseThe inequalities $0\le 4<9$ and the relation $13=1\cdot 9+4$ hold. At cell $(1,4)$, the hook length is $12-4+1=9=a{p}^k$.

For the partition of an integer $(12,1)$, the Hook-length formula giveswhich is divisible by $3$. This also agrees with the degree of the standard representation of the symmetric group $S_{13}$.

The obstruction is visible in the core of a partition: removing the horizontal rim hook of length nine leaves $\gamma =(3,1)$, which has no hook length nine and hence is the $9$-core. Its degree is ${\chi }^{(3,1)}(1)=3$. The large hook ensures the required weight of a partition here, but the core degree is not coprime to $p$.

The equivalence is true. Write $c_r(\eta )$ for the total size at level $r$ of the core tower of a partition $\eta$, and $q_r(\eta )$ for the corresponding total size in its quotient tower of a partition. We use the size relation and the P-adic valuation of a symmetric-group character degree from the core tower:Here $d_p$ is the base-p digit sum. These imply the character degree coprime to p from the core tower criterion: if $|\eta |={\sum }_r{\alpha }_r{p}^r$, then $p\nmid {\chi }^{\eta }(1)$ exactly when $c_r(\eta )={\alpha }_r$ at every level. Indeed carrying $p$ units from one position to the next decreases the sum of coefficients by $p-1$; the valuation vanishes precisely when there are no carries.

For $k\ge 1$, we also use power-core truncation of a prime-core tower: for $\gamma =C_{p^k}(\lambda )$, its core tower agrees with that of $\lambda$ below level $k$ and is empty at every level at least $k$. Moreover $q_k(\lambda )=w_{p^k}(\lambda )$, the weight of a partition for modulus $p^k$. To see the truncation, removal of a rim hook of length $p^k$ preserves the first $p$-core and, under the abacus divisible-hook correspondence, becomes removal of a $p^{k-1}$-hook in one quotient component. Iterating preserves all the lower cores; after all such removals, quotient level $k$ is empty.

If $p\nmid {\chi }^{\lambda }(1)$, the criterion gives $c_k(\lambda )=a$, $c_r(\lambda )=0$ for $r>k$, and the lower $c_r(\lambda )$ are the base-$p$ digits of $m$. It follows that $|\gamma |=m$, $p\nmid {\chi }^{\gamma }(1)$ andConversely, suppose $w_{p^k}(\lambda )=a$ and $p\nmid {\chi }^{\gamma }(1)$. Then $|\gamma |=m$ and its lower core-tower sizes are the digits of $m$. Quotient level $k$ of $\lambda$ has total size $a<p$. Each component there has size less than $p$, hence is already a $p$-core and has empty $p$-quotient. Thus $c_k(\lambda )=a$ and every higher level is empty. Combined with truncation, this gives exactly the digits of $n$, so $p\nmid {\chi }^{\lambda }(1)$.

ThereforeIf $k=0$, necessarily $m=0$ and $n=a<p$; the $1$-core is empty, its weight is $n=a$, and the Hook-length formula shows that every degree is coprime to $p$. Thus the equivalence holds in that boundary case too, with the empty partition having degree one.

The Murnaghan–Nakayama rule states that if a permutation has a $k$-cycle and remaining cycle type $\rho$, thenwhere $H$ ranges over removable rim hooks of length $k$ and $\mathrm{leg}(H)$ is one less than the number of rows occupied by $H$.

The Murnaghan–Nakayama rule states that if a permutation has a $k$-cycle and remaining cycle type $\rho$, thenwhere $H$ runs over the removable rim hooks of length $k$. Iterating removes rim hooks whose lengths are the cycle lengths, and the character value is the signed sum over all complete removal sequences.

For a prime number $p$, integer $k\ge 1$, and $\gamma =C_{p^k}(\lambda )$, taking the $p^k$-core of a partition preserves the entries of the core tower of a partition at levels $0,\dots ,k-1$ and makes every level at least $k$ empty. Also the weight of a partition $w_{p^k}(\lambda )$ equals the total size at level $k$ of the quotient tower of a partition.

For the first claim, a removable rim hook of length $p^k$ is a bead move by $p^k$ on the abacus of a partition. It preserves the numbers of beads on the $p$ runners and hence the $p$-core. Under the abacus divisible-hook correspondence, it becomes removal of a $p^{k-1}$-hook in one quotient component; iteration preserves the cores at the next $k-1$ levels. After all $p^k$-hooks are removed, every component at quotient level $k$ is empty, so every higher core level is empty as well. The weight assertion follows from iterated quotient equals a power quotient up to permutation and the size formula for the quotient of a partition.

Multiplication of an alternant by $p_r$ is the sum of alternants obtained by raising one row exponent by $r$. Equal exponents give zero. Otherwise sort the shifted beta number of a partition back into decreasing order. The resulting partition adds a size-$r$ rim hook; the number of crossed exponents is its height and supplies the determinant sign. This proves the formula through the bialternant formula. Iteration counts signed border-strip tableaux.

For diagram inclusion $\mu \subseteq \lambda$, the skew Young diagram $\lambda /\mu$ consists of the cells of $\lambda$ outside $\mu$. Diagram inclusion is not the dominance order on partitions. Edge adjacency defines its connected components, and a connected skew diagram without a $2\times 2$ square is a rim hook.

With $m$ beta numbers, the staircase has set $\{0,2,\dots ,2m-2\}$. Lowering a bead by a positive even number either hits an occupied bead or a negative position, so no even-size rim hook can be removed. The adjoint of multiplication by $p_r$ under the Hall inner product of symmetric functions is $r\partial /\partial p_r$. The power-sum border-strip multiplication formula therefore gives zero derivative with respect to every even power sum. Since the rational symmetric-function algebra is freely generated by the power sums, the stated conclusion follows.