Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.)

**(a)** ([Etingof's *Lectures on Representation Theory*][1], proof of Lemma 4.40): If $A$ and $B$ share the same Young diagram, then there exist two entries which lie in the same row of $A$ and in the same column of $B$, unless we can make $A$ and $B$ equal by permuting some elements inside their rows in $A$ and permuting some elements inside their columns in $B$.

**(b)** ([Etingof's *Lectures on Representation Theory*][2], proof of Lemma 4.41): If the partition corresponding to $A$ is lexicographically larger than that corresponding to $B$ - but both have the same sum -, then there exist two entries which lie in the same row of $A$ and in the same column of $B$.

**(c)** ([Serganova's Representation Notes][3], lecture 6, Lemma 1.4): If the Young diagram of $B$ has only one less square than $A$ but does not result from $A$ by removing one square, then there exist two entries which lie in the same row of $A$ and in the same column of $B$, or two entries which lie in the same row of $B$ and in the same column of $A$.

I am rather new to Young tableaux, and I haven't looked into Fulton, Stanley or Knuth, but maybe someone can answer on the spot whether there is a more general statement behind these three results?

Oh, and since this fits so nicely: [This paper][4] gives a wonderful proof of the Littlewood-Richardson rule, even generalized to the product of a Schur function of a Young diagram with that of a skew Young diagram. Is there a reasonable generalization to the product of two skew Young diagrams?

**UPDATE:** Claim **(c)** is wrong, as easily checked for $\lambda = \begin{array}{ccc} 5&1&3\\\\ 2& & \\\\ 4 & & \\\\ \end{array}$ and $\mu = \begin{array}{cc} 1& 2 \\\\ 4 & 3 \end{array}$.

  [1]: http://www-math.mit.edu/~etingof/replect.pdf
  [2]: http://www-math.mit.edu/~etingof/replect.pdf
  [3]: http://math.berkeley.edu/~serganov/math252/index.html
  [4]: http://www.combinatorics.org/Volume_9/PDF/v9i1n5.pdf