Set partitions and partitions without singleton blocks of type B

In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely.

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Actions and Identities on Set Partitions

The Electronic Journal of Combinatorics ◽

10.37236/1992 ◽

2012 ◽

Vol 19 (1) ◽

Cited By ~ 1

Author(s):

Eric Marberg

Keyword(s):

Abelian Group ◽

Group Action ◽

Type B ◽

Set Partition ◽

Set Partitions ◽

Unitriangular Group

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $\mathbb{A}$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $\mathbb{A}^n$ on the set of $\mathbb{A}$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning André and Neto's supercharacter theories of type B and D.

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Minimally Intersecting Set Partitions of Type $B$

The Electronic Journal of Combinatorics ◽

10.37236/294 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 1

Author(s):

William Y.C. Chen ◽

David G.L. Wang

Keyword(s):

Type B ◽

Set Partitions

Motivated by Pittel's study of minimally intersecting set partitions, we investigate minimally intersecting set partitions of type $B$. Our main result is a formula for the number of minimally intersecting $r$-tuples of $B_n$-partitions. As a consequence, it implies the formula of Benoumhani for the Dowling number in analogy to Dobiński's formula.

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On colored set partitions of type B n

Open Mathematics ◽

10.2478/s11533-014-0419-9 ◽

2014 ◽

Vol 12 (9) ◽

Author(s):

David Wang

Keyword(s):

Generating Function ◽

Asymptotic Expression ◽

Type B ◽

Set Partitions ◽

Asymptotic Normal

AbstractGeneralizing Reiner’s notion of set partitions of type B n, we define colored B n-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored B n-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored B n-partition. We find an asymptotic expression of the total number of colored B n-partitions up to an error of O(n −1/2log7/2 n], and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored B n-partitions.

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Crossings and nestings in set partitions of classical types

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2826 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Martin Rubey ◽

Christian Stump

Keyword(s):

Nous Avons ◽

Type A ◽

The Other ◽

Type B ◽

Set Partition ◽

Set Partitions ◽

Type D ◽

Nous Montrons ◽

International Audience ◽

The One

International audience In this extended abstract, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely. Dans ce résumé, nous étudions des bijections entre diverses classes de partitions d'ensemble de types classiques qui préservent les "openers'' et les "closers''. D'une part, nous présentons des bijections pour les types $B$ et $C$ qui échangent croisées et emboôtées, qui généralisent une construction de Kasraoui et Zeng pour le type $A$. D'autre part, nous généralisons une bijection pour le type $B$ et $C$ qui échange la cardinalité d'un croisement maximal avec la cardinalité d'un emboîtement maximal comme il a été fait par Chen, Deng, Du, Stanley et Yan pour le type $A$. Pour le type $D$, nous avons seulement construit une bijection entre les partitions non croisées et non emboîtées. Pour tout les types classiques, nous montrons que l'ensemble des "openers'' et l'ensemble des "closers'' déterminent une partition non croisées ou non emboîtées essentiellement de façon unique.

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Biological and Morphological Studies on Low-Leukemia-Strain Mice with Hodgkin's-Like Neoplasia Induced by Serial Cell-Free Transmission of Leukemic Organs of SJL/J Strain Mice

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100100792 ◽

1968 ◽

Vol 26 ◽

pp. 210-211

Author(s):

S. Fujinaga ◽

K. Maruyama ◽

C.W. Williams ◽

K. Sekhri ◽

L. Dmochowski

Keyword(s):

Tissue Culture ◽

Type A ◽

Reticulum Cell ◽

Type B ◽

Viral Origin ◽

Serial Transfer ◽

Strain Mouse ◽

Morphological Studies ◽

Cell Neoplasm ◽

Free Material

Yumoto and Dmochowski (Cancer Res.27, 2098 (1967)) reported the presence of mature and immature type C leukemia virus particles in leukemic organs and tissues such as lymph nodes, spleen, thymus, liver, and kidneys of SJL/J strain mice with Hodgki's-like disease or reticulum cell neoplasm (type B). In an attempt to ascertain the possibility that this neoplasia may be of viral origin, experiments with induction and transmission of this neoplasm were carried out using cell-free extracts of leukemic organs from an SJL/J strain mouse with spontaneous disease.It has been possible to induce the disease in low-leukemia BALB/c and C3HZB strain mice and serially transfer the neoplasia by cell-free extracts of leukemic organs of these mice. Histological examination revealed the neoplasia to be of either reticulum cell-type A or type B. Serial transfer is now in its fifth passage. In addition leukemic spleen from another SJL/J strain mouse with spontaneous reticulum cell neoplasm (type A) was set up in tissue culture and is now in its 141st serial passage in vitro. Preliminary results indicate that cell-free material of 39th tissue culture passage can reproduce neoplasia in BALB/c mice.

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Cryo-electron microscopy of two-dimensional crystals of reduced type B botulinum neurotoxin

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100123052 ◽

1992 ◽

Vol 50 (1) ◽

pp. 530-531

Author(s):

P. F. Flicker ◽

V.S. Kulkarni ◽

J. P. Robinson ◽

G. Stubbs ◽

B. R. DasGupta

Keyword(s):

Disulfide Bonds ◽

Clostridium Botulinum ◽

Structural Changes ◽

Two Dimensional ◽

Type B ◽

Single Chain ◽

Muscle Paralysis ◽

Cryo Electron Microscopy ◽

Disulfide Linkages ◽

Intracellular Action

Botulinum toxin is a potent neurotoxin produced by Clostridium botulinum. The toxin inhibits release of neurotransmitter, causing muscle paralysis. There are several serotypes, A to G, all of molecular weight about 150,000. The protein exists as a single chain or or as two chains, with two disulfide linkages. In a recent investigation on intracellular action of neurotoxins it was reported that type B neurotoxin can inhibit the release of Ca++-activated [3H] norepinephrine only if the disulfide bonds are reduced. In order to investigate possible structural changes in the toxin upon reduction of the disulfide bonds, we have prepared two-dimensional crystals of reduced type B neurotoxin. These two-dimensional crystals will be compared with those of the native (unreduced) type B toxin.

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