Wilf’s conjecture in fixed multiplicity

We give an algorithm to determine whether Wilf’s conjecture holds for all numerical semigroups with a given multiplicity [Formula: see text], and use it to prove Wilf’s conjecture holds whenever [Formula: see text]. Our algorithm utilizes techniques from polyhedral geometry, and includes a parallelizable algorithm for enumerating the faces of any polyhedral cone up to orbits of an automorphism group. We also introduce a new method of verifying Wilf’s conjecture via a combinatorially flavored game played on the elements of a certain finite poset.

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A Graph-Theoretic Approach to Wilf's Conjecture

The Electronic Journal of Combinatorics ◽

10.37236/9106 ◽

2020 ◽

Vol 27 (2) ◽

Cited By ~ 1

Author(s):

Shalom Eliahou

Keyword(s):

Graph Theory ◽

Numerical Semigroup ◽

Numerical Semigroups ◽

Theoretic Approach ◽

Primitive Elements ◽

Graph Theoretic ◽

Graph Theoretic Approach ◽

Wilf’S Conjecture ◽

Open Question

Let $S \subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m = \min(S \setminus \{0\})$ and conductor $c=\max(\mathbb{N} \setminus S)+1$. Let $P$ be the set of primitive elements of $S$, and let $L$ be the set of elements of $S$ which are smaller than $c$. A longstanding open question by Wilf in 1978 asks whether the inequality $|P||L| \ge c$ always holds. Among many partial results, Wilf's conjecture has been shown to hold in case $|P| \ge m/2$ by Sammartano in 2012. Using graph theory in an essential way, we extend the verification of Wilf's conjecture to the case $|P| \ge m/3$. This case covers more than $99.999\%$ of numerical semigroups of genus $g \le 45$.

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Wilf’s conjecture for numerical semigroups with large second generator

Journal of Algebra and Its Applications ◽

10.1142/s0219498821501978 ◽

2020 ◽

pp. 2150197

Author(s):

Dario Spirito

Keyword(s):

Quadratic Function ◽

Numerical Semigroups ◽

Embedding Dimension ◽

Wilf’S Conjecture

We study Wilf’s conjecture for numerical semigroups [Formula: see text] such that the second least generator [Formula: see text] of [Formula: see text] satisfies [Formula: see text], where [Formula: see text] is the conductor and [Formula: see text] the multiplicity of [Formula: see text]. In particular, we show that for these semigroups Wilf’s conjecture holds when the multiplicity is bounded by a quadratic function of the embedding dimension.

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Numerical semigroups with large embedding dimension satisfy Wilf’s conjecture

Semigroup Forum ◽

10.1007/s00233-011-9370-2 ◽

2012 ◽

Vol 85 (3) ◽

pp. 439-447 ◽

Cited By ~ 9

Author(s):

Alessio Sammartano

Keyword(s):

Numerical Semigroups ◽

Embedding Dimension ◽

Wilf’S Conjecture

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A generalization of Wilf’s conjecture for generalized numerical semigroups

Semigroup Forum ◽

10.1007/s00233-020-10085-7 ◽

2020 ◽

Vol 101 (2) ◽

pp. 303-325

Author(s):

Carmelo Cisto ◽

Michael DiPasquale ◽

Gioia Failla ◽

Zachary Flores ◽

Chris Peterson ◽

...

Keyword(s):

Numerical Semigroups ◽

Wilf’S Conjecture

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On a special case of Wilf’s conjecture

Asian-European Journal of Mathematics ◽

10.1142/s1793557120501594 ◽

2020 ◽

Vol 13 (08) ◽

pp. 2050159

Author(s):

Violeta Angjelkoska ◽

Donco Dimovski

Keyword(s):

Numerical Semigroup ◽

Frobenius Number ◽

Embedding Dimension ◽

Minimal Set ◽

Dimension Formula ◽

Wilf’S Conjecture ◽

Number Formula ◽

Genus Formula ◽

Multiplicity Formula ◽

Special Case

Let [Formula: see text] be a numerical semigroup with embedding dimension [Formula: see text], minimal set of generators [Formula: see text], Frobenius number [Formula: see text], multiplicity [Formula: see text] and genus [Formula: see text]. In this paper, we prove that Wilfs conjecture i.e. the inequality [Formula: see text] holds for [Formula: see text] when [Formula: see text] is a basis for [Formula: see text]

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Selective Sampling and Orientation of Non-Homogeneous Tissues

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100100238 ◽

1968 ◽

Vol 26 ◽

pp. 98-99

Author(s):

C. C. Clawson ◽

L. W. Anderson ◽

R. A. Good

Keyword(s):

Electron Microscopy ◽

Electron Microscope ◽

Light Microscopy ◽

Random Sampling ◽

New Method ◽

Microscope Examination ◽

Small Areas ◽

Selective Sampling ◽

Large Numbers ◽

Specific Orientation

Investigations which require electron microscope examination of a few specific areas of non-homogeneous tissues make random sampling of small blocks an inefficient and unrewarding procedure. Therefore, several investigators have devised methods which allow obtaining sample blocks for electron microscopy from region of tissue previously identified by light microscopy of present here techniques which make possible: 1) sampling tissue for electron microscopy from selected areas previously identified by light microscopy of relatively large pieces of tissue; 2) dehydration and embedding large numbers of individually identified blocks while keeping each one separate; 3) a new method of maintaining specific orientation of blocks during embedding; 4) special light microscopic staining or fluorescent procedures and electron microscopy on immediately adjacent small areas of tissue.

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