Asymptotic prime-power divisibility of binomial, generalized binomial, and multinomial coefficients

Holte, John M.

doi:10.1090/s0002-9947-97-01794-7

Cited by 8 publications

(5 citation statements)

References 63 publications

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“…(note i is never even in the first three sums, since then 2 t + 1 i is false; this justifies the second last equality, since in the last sum j runs through disjoint pairs of consecutive integers) where the last sum is wt(X(2 t , 2 t+1 ℓ − 1) by (12) and so is 2 n−2 by Theorem 4. Thus we have proved that (11) implies (13).…”

Section: Now We Determine Whenmentioning

confidence: 55%

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Balanced Symmetric Functions Over ${\hbox{GF}}(p)$

Cusick

Stănică

2008

IEEE Trans. Inform. Theory

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Under mild conditions on n, p, we give a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF (p), where p is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that X(2 t , 2 t+1 l − 1) are the only nonlinear balanced elementary symmetric polynomials over GF (2), whereand we prove various results in support of this conjecture.

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Section: Now We Determine Whenmentioning

confidence: 55%

“…Proof. Assume m i < p. By a known extension of Kummer's result that belongs to Dickson (see [13,Theorem D,p. 3860]) the power of p that divides the multinomial coefficient equals the number of carries when we add m 0 + m 1 + • • • + m n in base p, but the mentioned sum is equal to p, therefore the number of carries is 1.…”

Section: Enumeration Resultsmentioning

confidence: 94%

Balanced Symmetric Functions Over ${\hbox{GF}}(p)$

Cusick

Stănică

2008

IEEE Trans. Inform. Theory

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“…The carries Markov chain is truly ubiquitous; see Chapter 6 of [6] for a survey. Its many appearances in mathematics include: riffle shuffling ( [4], [5]), fractals [9], additive combinatorics [7], and cohomology [10].…”

Section: The Proof Of Theorem 21 Of This Paper Shows Thatmentioning

confidence: 99%

Card shuffling and P-partitions

Fulman

Petersen

2021

Discrete Mathematics

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“…Using a theorem of E. Kummer [9] (see also [7]), we obtain the following explicit description of the rational G a -submodule G a • T n . Proposition 1.5.…”

Section: The Colimitmentioning

confidence: 99%

Filtrations, 1-parameter subgroups, and rational injectivity

Friedlander

2018

Advances in Mathematics

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Abstract. We investigate rational G-modules M for a linear algebraic group G over an algebraically closed field k of characteristic p > 0 using filtrations by sub-coalgebras of the coordinate algebra k[G] of G. Even in the special case of the additive group Ga, interesting structures and examples are revealed. The "degree" filtration we consider for unipotent algebraic groups leads to a "filtration by exponential degree" applicable to rational G modules for any linear algebraic group G of exponential type; this filtration is defined in terms of 1-parameter subgroups and is related to support varieties introduced recently by the author for such rational G-modules. We formulate in terms of this filtration a necessary and sufficient condition for rational injectivity for rational G-modules. Our investigation leads to the consideration of two new classes of rational G-modules: those that are "mock injective" and those that are "mock trivial". IntroductionBeginning with the very special case of the additive group G a , we consider the filtration by degree on rational G a -modules which enables us to better understand the intriguing category (G a -M od) of rational G a -modules. This filtration leads to a similarly defined filtration by degree on rational U N -modules, where U N ⊂ GL N is the closed subgroup of strictly upper triangular matrices, and determines a filtration on rational U -modules for a closed linear subgroup U ⊂ U N . We then initiate the study of a less evident filtration on rational G-modules for G a linear algebraic group of exponential type. For rational U N -modules for the unipotent algebraic group U N , this filtration of exponential degree is a comparable to the more elementary filtration by degree we first consider. Throughout, we fix an algebraic closed field k of characteristic p > 0 and consider (smooth) linear algebraic groups over k together with their rational actions on k-vector spaces.In some sense, this paper is a sequel to the author's recent paper [4] in which a theory of support varieties M → V (G) M was constructed for rational G-modules. The construction of the filtration by exponential degreeuses restrictions of M to 1-parameter subgroups G a → G, and thus is based upon actions of G on M at p-unipotent elements of G. The role of 1-parameter subgroups to study rational G-modules was introduced in [3]; the property of p-unipotent degree introduced in [3, 2.5] is the precursor to our filtration by exponential degree. The origins of this approach to filtrations lie in considerations of support varieties

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Asymptotic prime-power divisibility of binomial, generalized binomial, and multinomial coefficients

Cited by 8 publications

References 63 publications

Balanced Symmetric Functions Over ${\hbox{GF}}(p)$

Balanced Symmetric Functions Over ${\hbox{GF}}(p)$

Card shuffling and P-partitions

Filtrations, 1-parameter subgroups, and rational injectivity

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