Ideals, representations and a symmetrised Bernoulli triangle

Abstract We study some representations of symmetric groups arising from a certain ideal in the coordinate ring of affine n -space. Our results give graded and representation-theoretic enhancements of the sequence of numbers $$c_n=1 + (n-2)\cdot 2^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>·</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , sequence 337 of the Online Encyclopaedia of Integer Sequences, involving a symmetric version of the Bernoulli triangle.