论文信息 - Symmetry in Maximal (s-1, s+1) Cores

Symmetry in Maximal (s-1, s+1) Cores

We explain a "curious symmetry" for maximal $(s-1,s+1)$-core partitions first observed by T. Amdeberhan and E. Leven. Specifically, using the $s$-abacus, we show such partitions have empty $s$-core and that their $s$-quotient is comprised of 2-cores. This imposes strong conditions on the partition structure, and implies both the Amdeberhan-Leven result and additional symmetry. We also find a more general family that exhibits these symmetries.

Rishi Nath

[1] Matthew Fayers. The t-core of an s-core , 2011, J. Comb. Theory, Ser. A.

[2] Jørn B. Olsson,et al. Combinatorics and representations of finite groups , 1993 .

[3] The number of self-conjugate core partitions , 2012, 1201.6629.

[4] Amol Aggarwal. Armstrong's conjecture for (k, mk+1)-core partitions , 2015, Eur. J. Comb..

[5] Amitabha Tripathi,et al. On the largest size of a partition that is both s-core and t-core , 2009 .

[6] Jaclyn Anderson,et al. Partitions which are simultaneously t1- and t2-core , 2002, Discret. Math..

[7] Dennis Stanton,et al. Block inclusions and cores of partitions , 2007 .

[8] Robin D. P. Zhou,et al. On the enumeration of (s, s+1, s+2)-core partitions , 2014, Eur. J. Comb..

[9] Tewodros Amdeberhan,et al. Multi-cores, posets, and lattice paths , 2014, Adv. Appl. Math..

[10] William Y.C. Chen,et al. Average Size of a Self-conjugate (s, t)-Core Partition , 2014 .

[11] Gordon James,et al. The Representation Theory of the Symmetric Groups , 1977 .

[12] Brittany Halverson-Duncan. The Representation Theory of the Symmetric Groups , 2014 .

[13] Huan Xiong,et al. On the largest size of (t, t+1, ..., t+p)-core partitions , 2014, Discret. Math..