# permutation

## Definitions

• WordNet 3.6
• n permutation act of changing the lineal order of objects in a group
• n permutation complete change in character or condition "the permutations...taking place in the physical world"- Henry Miller"
• n permutation the act of changing the arrangement of a given number of elements
• n permutation an event in which one thing is substituted for another "the replacement of lost blood by a transfusion of donor blood"
• ***
Webster's Revised Unabridged Dictionary
• Permutation (Math) Any one of such possible arrangements.
• Permutation (Law) Barter; exchange.
• Permutation The act of permuting; exchange of the thing for another; mutual transference; interchange. "The violent convulsions and permutations that have been made in property."
• Permutation (Math) The arrangement of any determinate number of things, as units, objects, letters, etc., in all possible orders, one after the other; -- called also alternation. Cf. Combination n., 4.
• ***
Century Dictionary and Cyclopedia
• n permutation Interchange; concurrent changes; mutual change; change in general.
• n permutation Exchange; barter.
• n permutation In mathematics, a linear arrangement of objects resulting from a change of their order. Permutation differs from combination in this, that in the latter there is no reference to the order in which the quantities are combined, whereas in the former this order is considered, and consequently the number of permutations always exceeds the number of combinations.If n represents the number of quantities, then the number of permutations that can be formed out of them, taking two by two together, is n X (n-1); taken three and three together, it is n X (n-1) X (n-2); and so on. Sometimes called alternation. See combination, 5.
• n permutation In philology, the mutation or interchange of consonants, especially of allied consonants.
• ***
Chambers's Twentieth Century Dictionary
• Permutation act of changing one thing for another:
• Permutation (math.) the arrangement of things or letters in every possible order
• ***

## Etymology

Webster's Revised Unabridged Dictionary
L. permutatio,: cf. F. permutation,. See Permute
Chambers's Twentieth Century Dictionary
L.,—permutāreper, through, mutāre, to change.

## Usage

### In literature:

There would still remain the inverse permutation: to obtain only females and no males, or very few.
"Bramble-bees and Others" by J. Henri Fabre
Pen in hand, I arrange, permute and combine.
"The Life of the Fly" by J. Henri Fabre
There would still remain the inverse permutation: to obtain only females and no males, or very few.
"The Wonders of Instinct" by J. H. Fabre
The permutations and combinations of the stores were multiplied.
"The River War" by Winston S. Churchill
We know only five such permuted alphabets, but there may be more.
"Hebraic Literature; Translations from the Talmud, Midrashim and Kabbala" by Various
He says that every possible permutation of the four bells must be rung once, and once only.
"Amusements in Mathematics" by Henry Ernest Dudeney
And though words, too, are finite, their permutations are infinite.
"The Continental Monthly, Vol. 2, No 3, September, 1862" by Various
He is not a permutation and combination of old elements, transferred through the parents.
"Fantasia of the Unconscious" by D. H. Lawrence
Calculations of the possible combinations and permutations of a small number of objects are familiar.
"The Classification of Patents" by United States Patent Office
What could he imagine, except combinations and permutations of his own memories?
"Suite Mentale" by Gordon Randall Garrett
In abortion it exists, but in a stunted and dwarfed, but not otherwise permuted state.
"Vegetable Teratology" by Maxwell T. Masters
The various permutations of the X-shaped scapular mark and dark sacral marks differ proportionately in different samples.
"Middle American Frogs of the Hyla microcephala Group" by William E. Duellman
This permutation was not quite unnatural; for Wilhelm and Laertes did resemble one another, though in a very distant manner.
"Wilhelm Meister's Apprenticeship and Travels, Vol. I (of 2)" by Johann Wolfgang von Goethe
Watch now the permutations and combinations the author develops from this general situation.
"Dramatic Technique" by George Pierce Baker
Those ten little Arabic symbols and their combinations and permutations held a fascination for him.
"The Sin of Monsieur Pettipon" by Richard Connell
We can do this in twenty-four different ways by forming all permutations of the letters.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
Then began fresh combinations and permutations of the talk.
"A Crooked Mile" by Oliver Onions
Any two translations are permutable with each other, and give when combined another translation.
"Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6" by Various
Expertness in these "permutations and combinations" is only to be arrived at by constant practice.
"Riding for Ladies" by W. A. Kerr
In genuine Lancashire we have first the old-accustomed permutations of the vowels.
"Lancashire" by Leo H. (Leo Hartley) Grindon
***

### In poetry:

French Deck. Solitary stroker of cards. He
dealt himself a hand. Turn stills of the past in
unending permutations, shuffle and begin. Sort
the images again. And sort them again. This
game reveals germs of truth, and death.
"Notes On Vision" by James Douglas Morrison

### In news:

As we talked about on Friday, the Knicks have gone through so many different permutations in this season alone that it's enough to give fans whiplash, or vertigo, or some combination of both.
One of the ISAM Live 2.0 set's many stunning permutations.
Cycles, and the cycle decomposition of a permutation.
The word and its permutations come up a lot when the singer-keyboardist-producer talks about Johnny Headband 's Who Cooks for You.
Permut Presentations presentation of a James W Evangelatos, Julie Lott Gallo production.
Produced by David Permut, Mark Sennet, Julie Lott Gallo, James W Evangelatos.
There is a brief but thorough description of what a combinatorial object is, such as permutations and subsets.
Combinations and permutations are fairly rare on the SAT.
Questions that involve arranging a group are " permutations ," since in an arrangement the order of things is important.
Now that we have the permutation fully specified, we'll use it to learn a little about how the CTC permutation works — in particular, we're going to learn about cycles in permutations and try to understand the cycle structure of a CTC.
Randomization was based on permuted blocks, with a block size of 4.
John Whitney's Permutations by Klax.
One of the first theorems on permutation routing, proved by V.E.
Columnar transposition ciphers and permutations, oh my.
Run a few permutations, described below, and Xenopohon ends up using the word "paradeisos," which he used to refer to the walled gardens of Persia.
***

### In science:

According to Section 10.2, B -permutations for the associahedron are 312-avoiding permutations and h-polynomial is equal to the sum hPB(Pathn ) (t) = Pw qpeak(w)−1 over all 312-avoiding permutations w ∈ Sn .
Faces of Generalized Permutohedra
Our goal here is therefore to present a simple bijection between permutations and permutation tableaux that maps the descents of the permutation to the columns of the tableau.
A simple bijection between permutation tableaux and permutations
Williams, Permutation tableaux and permutation patterns, to appear in the Journal of Combinatorial Theory, Series A. arXiv:math.
A simple bijection between permutation tableaux and permutations
We determine exactly the probability of ﬁnding the tra jectory of such a permutation-generated random walk at site X at time n, obtain the probability measure of different excursions and deﬁne the asymptotic distribution of the number of ”U-turns” of the tra jectories - permutation ”peaks” and ”through”.
Random patterns generated by random permutations of natural numbers
First we draw uniformly a permutation and after that we pick N initial points which are permuted according to the chosen permutation to obtain N terminal points.
Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks
This is seen in the deﬁnition (1.1) of the symmetrised measure P(sym) N ,β , where permutations are sampled uniformly and for each permutation there is a product of not necessarily identical distributions of single random walks with initial and terminal condition.
Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks
Permuting ¯i′ we get an element ˜¯i ∈ Cn and we will denote the sign of this permutation by γ¯i .
Fusion algebras for imprimitive complex reflection groups
Here, d(λ) is the number of permutations with rec(πn ) = (rk ) and (rk ) corresponding to λ via (8), and d(λ, µ) is the number of permutations πm which correspond to µ and are coherent with some ﬁxed permutation πn having this rec(πn ) = (rk ).
Coherent random permutations with record statistics
Under the commutativity property of multiplication, the product XtXt+xXt+y remains the same under any permutation of the three variables, so for each permutations in S3 , we permute the variables, according to the permutation, then adjust t so that the ﬁrst variable has index t.
Multivariate Lag-Windows and Group Representations
When the coordinates of b1 and b2 are permuted by a permutation in S3 , the resulting vector still lies in V since the coordinates still sum to 0.
Multivariate Lag-Windows and Group Representations
If we ﬁx any permutation in S3 and multiply it with all the elements of S3 , the resulting action just permutes the six elements of S3 ; this is a type of group action where S3 “acts on itself ”.
Multivariate Lag-Windows and Group Representations
It is well known that any permutation can be written as a product of cycles and every cycle is a product of transpositions (2-cycles), therefore the permutations can be generated by just the transpositions.
Multivariate Lag-Windows and Group Representations
One easy way to prove this is to show that the only permutation being mapped to hI i is e, the identity permutation.
Multivariate Lag-Windows and Group Representations
If we consider the permutations that ﬁx the ﬁrst label, then we see that the identities in (17) induced by these (k − 1)! permutations are just permutations of the arguments.
Multivariate Lag-Windows and Group Representations
Hence for each k , the number of permutations in Sn with k inversions is the same as the number of permutations in Sn whose rightward distance is k .
Random and exhaustive generation of permutations and cycles
***