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+#+title: A \(q\)-weighted Robinson-Schensted algorithm
+
+#+date: <2013-06-01>
+
+In [[https://projecteuclid.org/euclid.ejp/1465064320][this paper]] with
+[[http://www.bristol.ac.uk/maths/people/neil-m-oconnell/][Neil]] we
+construct a \(q\)-version of the Robinson-Schensted algorithm with
+column insertion. Like the
+[[http://en.wikipedia.org/wiki/Robinson–Schensted_correspondence][usual
+RS correspondence]] with column insertion, this algorithm could take
+words as input. Unlike the usual RS algorithm, the output is a set of
+weighted pairs of semistandard and standard Young tableaux \((P,Q)\)
+with the same shape. The weights are rational functions of indeterminant
+\(q\).
+
+If \(q\in[0,1]\), the algorithm can be considered as a randomised RS
+algorithm, with 0 and 1 being two interesting cases. When \(q\to0\), it
+is reduced to the latter usual RS algorithm; while when \(q\to1\) with
+proper scaling it should scale to directed random polymer model in
+[[http://arxiv.org/abs/0910.0069][(O'Connell 2012)]]. When the input
+word \(w\) is a random walk:
+
+\begin{align*}\mathbb
+P(w=v)=\prod_{i=1}^na_{v_i},\qquad\sum_ja_j=1\end{align*}
+
+the shape of output evolves as a Markov chain with kernel related to
+\(q\)-Whittaker functions, which are Macdonald functions when \(t=0\)
+with a factor.