= Staircase Schur functions use only odd power sums
{title2=$s_{(m-1,m-2,\ldots,1)}\in\mathbb Q[p_1,p_3,\ldots]$}
With $m$ beta numbers, the staircase has set $\{0,2,\ldots,2m-2\}$. 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.
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