Combining results of T.K. Lam and J. Stembridge, the type C Stanley symmetric function F C w (x), indexed by an element w in the type C Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements. . the connections between our crystal operators and those obtained by intertwining crystal operators on words with Haiman's symmetrization of shifted mixed insertion [8, Section 5] and the conversion map [17, Proposition 14] as outlined in Remark 4.11. We thank Toya Hiroshima for pointing out that the definition of

The notion of (a, b)-cores is closely related to rational (a, b) Dyck paths due to Anderson's bijection, and thus the number of (a, a + 1)-cores is given by the Catalan number C a . Recent research shows that (a, a + 1) cores with distinct parts are enumerated by another important sequence-Fibonacci numbers F a . In this paper, we consider the abacus description of (a, b)-cores to introduce the natural grading and generalize this result to (a, as + 1)-cores. We also use the bijection with Dyck paths to count the number of (2k − 1, 2k + 1)-cores with distinct parts. We give a second grading to Fibonacci numbers, induced by bigraded Catalan sequence C a,b (q, t).Part (1) of the above theorem was independently proved by A.Straub [13]. Another interesting conjecture of Amdeberhan is the number of (2k − 1, 2k + 1)-cores with distinct parts. This conjecture have been proven by Yan, Qin, Jin and Zhou [15]:Theorem 2. (YQJZ,16) The number of (2k − 1, 2k + 1)-cores with distinct parts is equal to 2 2k−2 . The proof uses somewhat complicated arguments about the poset structure of cores. Results by Zaleski and Zeilberger [17] improve the argument using Experimental Mathematics tools in Maple. More recently Baek, Nam and Yu provided simpler bijective proof in [6]. 1 arXiv:1705.09991v1 [math.CO]

Exploratory analysis over network data is often limited by our ability to efficiently calculate graph statistics, which can provide a model-free understanding of macroscopic properties of a network. This work introduces a framework for estimating the graphlet count-the number of occurrences of a small subgraph motif (e.g. a wedge or a triangle) in the network. For massive graphs, where accessing the whole graph is not possible, the only viable algorithms are those which act locally by making a limited number of vertex neighborhood queries. We introduce a Monte Carlo sampling technique for graphlet counts, called lifting, which can simultaneously sample all graphlets of size up to k vertices. We outline three variants of lifted graphlet counts: the ordered, unordered, and shotgun estimators. We prove that our graphlet count updates are unbiased for the true graphlet count, have low correlation between samples, and have a controlled variance. We compare the experimental performance of lifted graphlet counts to the state-of-the art graphlet sampling procedures: Waddling and the pairwise subgraph random walk.

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