Set-partition tableaux and representations of diagram algebras

In this paper, both software model visualization with path simulation and associated machining product are produced based on the step ring based 3-axis path planning to demo model-driven graphics processing unit (GPU) feature in tool path planning and 3D image model classification by GPU simulation. Subtractive 3D printing (i.e., 3D machining) is represented as integration between 3D printing modeling and CNC machining via GPU simulated software. Path planning is applied through material surface removal visualization in high resolution and 3D path simulation via ring selective path planning based on accessibility of path through pattern selection. First, the step ring selects critical features to reconstruct computer aided design (CAD) design model as STL (stereolithography) voxel, and then local optimization is attained within interested ring area for time and energy saving of GPU volume generation as compared to global all automatic path planning with longer latency. The reconstructed CAD model comes from an original sample (GATech buzz) with 2D image information. CAD model for optimization and validation is adopted to sustain manufacturing reproduction based on system simulation feedback. To avoid collision with the produced path from retraction path, we pick adaptive ring path generation and prediction in each planning iteration, which may also minimize material removal. Moreover, we did partition analysis and g-code optimization for large scale model and high density volume data. Image classification and grid analysis based on adaptive 3D tree depth are proposed for multi-level set partition of the model to define no cutting zones. After that, accessibility map is computed based on accessibility space for rotational angular space of path orientation to compare step ring based pass planning verses global all path planning. Feature analysis via central processing unit (CPU) or GPU processor for GPU map computation contributes to high performance computing and cloud computing potential through parallel computing application of subtractive 3D printing in the future.

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Asymptotics and Random Set Partition

Combinatorics of Set Partitions ◽

10.1201/b12691-12 ◽

2012 ◽

pp. 407-450

Keyword(s):

Random Set ◽

Set Partition

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On Pattern-Avoiding Partitions

The Electronic Journal of Combinatorics ◽

10.37236/763 ◽

2008 ◽

Vol 15 (1) ◽

Cited By ~ 13

Author(s):

Vít Jelínek ◽

Toufik Mansour

Keyword(s):

Stirling Numbers ◽

Equivalence Classes ◽

Catalan Numbers ◽

Set Partition ◽

Combinatorial Interpretations

A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$. We choose the ordering of the blocks so that they satisfy $\min B_1 < \min B_2 < \cdots < \min B_d$. We represent such a set partition by a canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, with $\pi_i=j$ if $i\in B_j$. We say that a partition $\pi$ contains a partition $\sigma$ if the canonical sequence of $\pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $\sigma$. Two partitions $\sigma$ and $\sigma'$ are equivalent, if there is a size-preserving bijection between $\sigma$-avoiding and $\sigma'$-avoiding partitions. We determine all the equivalence classes of partitions of size at most $7$. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most $3$. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

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PARAMETER SET PARTITION ACQUISITION BASED ON INFORMATION INDEX OF DEGREE OF RELATIONSHIP

Bulletin of NTU KhPI Series Strategic Management Portfolio Program and Project Management ◽

10.20998/2413-3000.2015.1111.9 ◽

2015 ◽

Vol 4 (2(1111)) ◽

pp. 46-51

Author(s):

Игорь Петрович ГАМАЮН ◽

Ольга Николаевна БЕЗМЕНОВА

Keyword(s):

Set Partition ◽

Information Index

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Statistical Distributions and $q$-Analogues of $k$-Fibonacci Numbers

The Electronic Journal of Combinatorics ◽

10.37236/2550 ◽

2013 ◽

Vol 20 (1) ◽

Author(s):

Adam M Goyt ◽

Brady L Keller ◽

Jonathan E Rue

Keyword(s):

Fibonacci Numbers ◽

Natural Extension ◽

Statistical Distributions ◽

Set Partition ◽

Set Partitions ◽

Partition Statistics

We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most k. The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics. We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers. This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations. In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles. We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities.

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