In mathematics, Monk's formula, found by , is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold.
Write t<sub>ij</sub> for the transposition (i j), and s<sub>i</sub> = t<sub>i,i+1</sub>. Then ðÂÂÂ<sub>s<sub>r</sub></sub> = x<sub>1</sub> + ⯠+ x<sub>r</sub>, and Monk's formula states that for a permutation w,
where is the length of w. The pairs (i, j) appearing in the sum are exactly those such that i ≤ r < j, w<sub>i</sub> < w<sub>j</sub>, and there is no i < k < j with w<sub>i</sub> < w<sub>k</sub> < w<sub>j</sub>; each wt<sub>ij</sub> is a cover of w in Bruhat order.