A bijection between irreducible kk-shapes and surjective pistols of height k−1k−1

This paper constructs a bijection between irreducible kk-shapes and surjective pistols of height k−1k−1, which carries the “free kk-sites” to the fixed points of surjective pistols. It confirms a conjecture of Hivert and Mallet (FPSAC 2011) that the number of irreducible kk-shapes is the Genocchi number G2kG2k, and, with precision, that the irreducible kk-shapes (with k≥2k≥2) generate the Gandhi polynomial Q2k−2(x)Q2k−2(x) with respect to the statistic of free kk-sites.