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Rapid growth in the amount of data that is electronically recorded as part of routine clinical operations has generated great interest in the use of Big Data methodologies to address clinical and research questions. These methods can efficiently analyze and deliver insights from high-volume, high-variety, and high-growth rate datasets generated across the(More)
In this paper, we focus on a “local property” of permutations: value-peak. A permutation σ has a value-peak σ(i) if σ(i − 1) < σ(i) > σ(i + 1) for some i ∈ [2, n − 1]. Define V P (σ) as the set of value-peaks of the permutation σ. For any S ⊆ [3, n], define V Pn(S) such that V P (σ) = S. Let Pn = {S | V Pn(S) 6= ∅}. we make the set Pn into a poset Pn by(More)
The symmetry of the joint distribution of the numbers of crossings and nestings of length 2 has been observed in many combinatorial structures, including permutations, matchings, set partitions, linked partitions, and certain families of graphs. These results have been unified in the larger context of enumeration of northeast and southeast chains of length(More)
The circular peak set of a permutation σ is the set {σ(i) | σ(i−1) < σ(i) > σ(i+1)}. In this paper, we focus on the enumeration problems for permutations by circular peak sets. Let cpn(S) denote the number of the permutations of order n which have the circular peak set S. For the case with |S| = 0, 1, 2, we derive the explicit formulas for cpn(S). We also(More)
In this paper, let Pn,n+k;≤n+k (resp. Pn;≤s) denote the set of parking functions α = (a1, · · · , an) of length n with n+k (respe. n)parking spaces satisfying 1 ≤ ai ≤ n+k (resp. 1 ≤ ai ≤ s) for all i. Let pn,n+k;≤n+k = |Pn,n+k;≤n+k| and pn;≤s = |Pn;≤s|. Let P l n;≤s denote the set of parking functions α = (a1, · · · , an) ∈ Pn;≤s such that a1 = l and pn;≤s(More)
In this paper, let P l n;≤s;k denote a set of k-flaw preference sets (a1, . . . , an) with n parking spaces satisfying that 1 ≤ ai ≤ s for any i and a1 = l and p l n;≤s;k = |P l n;≤s;k|. We use a combinatorial approach to the enumeration of k-flaw preference sets by their leading terms. The approach relies on bijections between the k-flaw preference sets(More)
W. Xu, B. Anderson, L. Auberbach, T. Averett, W. Bertozzi, T. Black, J. Calarco, L. Cardman, G. D. Cates, Z. W. Chai, J. P. Chen, S. Choi, E. Chudakov, S. Churchwell, G. S. Corrado, C. Crawford, D. Dale, A. Deur, P. Djawotho, T. W. Donnelly, D. Dutta, J. M. Finn, H. Gao, R. Gilman, A. V. Glamazdin, C. Glashausser, W. Glöckle, J. Golak, J. Gomez, V. G.(More)
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