Tootfinder

Equal knapsack identities between symmetric group character degrees
David J. Hemmer, Armin Straub, Karlee J. Westrem
https://arxiv.org/abs/2510.00301 https://

Equal knapsack identities between symmetric group character degrees
We prove a series of ``knapsack'' type equalities for irreducible character degrees of symmetric groups. That is, we find disjoint subsets of the partitions of $n$ so that the two corresponding character-degree sums are equal. Our main result refines our recent description of the Riordan numbers as the sum of all character degrees $f^λ$ where $λ$ is a partition of $n$ into three parts of the same parity. In particular, the sum of the ``fat-hook'' degrees $f^{(k,k,1^{n-2k})}+f^{(k+1,k+1,1^{n-2…

On random bipartite graphs evolving by degrees
Neeladri Maitra
https://arxiv.org/abs/2509.23943 https://arxiv.org/pdf/2509.23943

On random bipartite graphs evolving by degrees
In this paper, we study a bipartite analogue of the `random graphs evolving by degrees' process. We are given a bipartitioned set of vertices $V$ into two disjoint parts ${L}$ and ${R}$ and possibly unequal positive constants $α$ and $β$. The graph evolves starting from $B_0$, the empty graph (with only isolated vertices). Given $B_t$, a non-adjacent vertex pair $u \in {L}, v \in {R}$ is sampled with probability proportional to $(d_u(t)+α)(d_v(t)+β)$, and the edge $\{u,v\}$ is included to $…