Instead, she immersed herself in the Freudian universe of deep, growling desires, her mind pitched at the ego's involutions and attachments.
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Her style has a sort of 18th-century gentility to it, pitched at a curious midpoint between Hume's clubbability and Kant's knotty involutions.
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Every corner we turned made R. gasp, every church we stepped into, every statue with its marble frothed up like surf, like the involutions of thought.
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I wanted this, for sure, but I was also seeking something else: to better understand my aging identity, not only in terms of my mind's involutions and attachments but no less crucially through the corporeal expressions of my organs, muscles, systems, and cells.
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In Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space Rn. Such involutions are easy to characterize and they can be described geometrically.
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Groups of symplectic type appear in centralizers of involutions of groups of GF(2)-type.
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Bender–Knuth involutions can be used to show that the number of semistandard skew tableaux of given shape and weight is unchanged under permutations of the weight. In turn this implies that the Schur function of a partition is a symmetric function. Bender–Knuth involutions were used by to give a short proof of the Littlewood–Richardson rule.
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There is another class of involutions, of trace 0, that move all 100 vertices.Wilson (2009), p. 213 As permutations in the alternating group A100, being products of an odd number (25) of double transpositions, these involutions lift to elements of order 4 in the double cover 2.A100. HS thus has a double cover 2.HS.
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For sets of size there are endofunctions on the set. Particular examples of bijective endofunctions are the involutions; i.e., the functions coinciding with their inverses.
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Every layered permutation is an involution. They are exactly the 231-avoiding involutions, and they are also exactly the 312-avoiding involutions. The layered permutations are a subset of the stack- sortable permutations, which forbid the pattern 231 but not the pattern 312. Like the stack-sortable permutations, they are also a subset of the separable permutations, the permutations formed by recursive combinations of direct and skew sums.
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Co0 has 4 conjugacy classes of involutions; these collapse to 2 in Co1, but there are 4-elements in Co0 that correspond to a third class of involutions in Co1. An image of a dodecad has a centralizer of type 211:M12:2, which is contained in a maximal subgroup of type 211:M24. An image of an octad or 16-set has a centralizer of the form 21+8.O8+(2), a maximal subgroup.
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In mathematical finite group theory, the Brauer–Fowler theorem, proved by , states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g1/3. The technique of the proof is to count involutions (elements of order 2) in G. Perhaps more important is another result that the authors derive from the same count of involutions, namely that up to isomorphism there are only a finite number of finite simple groups with a given centralizer of an involution. This suggested that finite simple groups could be classified by studying their centralizers of involutions, and it led to the discovery of several sporadic groups. Later it motivated a part of the classification of finite simple groups.
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In mathematical finite group theory, a group is said to be of characteristic 2 type or even type or of even characteristic if it resembles a group of Lie type over a field of characteristic 2. In the classification of finite simple groups, there is a major division between group of characteristic 2 type, where involutions resemble unipotent elements, and other groups, where involutions resemble semisimple elements. Groups of characteristic 2 type and rank at least 3 are classified by the trichotomy theorem.
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In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson , gives a formula for the order of a finite group in terms of the centralizers of involutions, extending the results of .
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The first proof that such a representation exists was given by Leonhard Euler in 1747 and was complicated. Since then, many different proofs have been found. Among them, the proof using Minkowski's theorem about convex setsSee Goldman's book, §22.5 and Don Zagier's short proof based on involutions have appeared.
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The projectivities are described algebraically as homographies, since the real numbers form a ring, according to the general construction of a projective line over a ring. Collectively they form the group PGL(2,R). The projectivities which are their own inverses are called involutions. A hyperbolic involution has two fixed points.
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The problem is to show that if a group has a centralizer of involution in "standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classical involution theorem. # Quasi-standard form # Central involutions # Classification of alternating groups. # Some sporadic groups # Thin groups.
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MULTI2 is a symmetric key algorithm with variable number of rounds. It has a block size of 64 bits, and a key size of 64 bits. A 256-bit implementation-dependent substitution box constant is used during key schedule. Scramble and descramble is done by repeating four basic functions (involutions).
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The number of involutions y(n) of a set with n elements is given by the recurrence equationy (n) = (n-1) \, y (n-2) + y (n-1).Applying for example Petkovšek's algorithm it is possible to see that there is no polynomial, rational or hypergeometric solution for this recurrence equation.
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Any involution in Co0 can be shown to be conjugate to an element of the Golay code. Co0 has 4 conjugacy classes of involutions. A permutation matrix of shape 212 can be shown to be conjugate to a dodecad. Its centralizer has the form 212:M12 and has conjugates inside the monomial subgroup.
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The textbook presentations in , , , , all contain various applications of the focal subgroup theorem relating fusion, transfer, and a certain kind of splitting called p-nilpotence. During the course of the Alperin–Brauer–Gorenstein theorem classifying finite simple groups with quasi- dihedral Sylow 2-subgroups, it becomes necessary to distinguish four types of groups with quasi-dihedral Sylow 2-subgroups: the 2-nilpotent groups, the Q-type groups whose focal subgroup is a generalized quaternion group of index 2, the D-type groups whose focal subgroup a dihedral group of index 2, and the QD-type groups whose focal subgroup is the entire quasi-dihedral group. In terms of fusion, the 2-nilpotent groups have 2 classes of involutions, and 2 classes of cyclic subgroups of order 4; the Q-type have 2 classes of involutions and one class of cyclic subgroup of order 4; the QD-type have one class each of involutions and cyclic subgroups of order 4. In other words, finite groups with quasi-dihedral Sylow 2-subgroups can be classified according to their focal subgroup, or equivalently, according to their fusion patterns.
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Suppose that W is a Coxeter group, generated by a set S of involutions, and P is a parabolic subgroup (the subgroup generated by some subset of S). A Coxeter matroid is a subset M of W/P that for every w in W, M contains a unique minimal element with respect to the w-Bruhat order.
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The triangles with reflection symmetry are isosceles, the quadrilaterals with this symmetry are kites and isosceles trapezoids. For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions for more), one of the three types of order two (involutions), hence algebraically isomorphic to C2. The fundamental domain is a half-plane or half-space.
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In 2 dimensions, a point reflection is a 180 degree rotation. Reflection symmetry can be generalized to other isometries of -dimensional space which are involutions, such as : in a certain system of Cartesian coordinates. This reflects the space along an -dimensional affine subspace. If = , then such a transformation is known as a point reflection, or an inversion through a point.
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Nathan Jacobson described the automorphisms of composition algebras in 1958. The classical composition algebras over and are unital algebras. Composition algebras without a multiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in The Book of Involutions, pp.
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With no way back, they join the cobbled- together cosmopolitan society. The story takes place in the Alliance-Union universe, in Union-side space. Cherryh originally named the story Involutions, because it "spirals in upon itself", but after discussing the title with her publisher, Wollheim's suggestion of Port Eternity was adopted.Author's introduction to Port Eternity in the Alternate Realities omnibus.
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In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two (up to isomorphism) extraspecial groups of order p1+2n. Extraspecial groups often occur in centralizers of involutions. The ordinary character theory of extraspecial groups is well understood.
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It is designed as a substitution–permutation network, which bears large similarity to Rijndael. Like KHAZAD, designed by the same authors and also submitted to NESSIE, it uses involutions for the various operations. An involution is an operation whose inverse is the same as the forward operation. In other words, when an involution is run twice, it is the same as performing no operation.
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In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity n-1 on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity n). The term inversion should not be confused with inversive geometry, where inversion is defined with respect to a circle.
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For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions), one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space. In certain contexts there is rotational as well as reflection symmetry. Then mirror-image symmetry is equivalent to inversion symmetry; in such contexts in modern physics the term parity or P-symmetry is used for both.
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This allows low-cost hardware and compact software implementations to use the same operations for both encryption and decryption. Both the S-box and the mix columns operations are involutions. Although many involutional components can make a cipher more susceptible to distinguishing attacks exploiting the cycle structure of permutations within the cipher, no attack strategy for the Anubis cipher has been presented. There are two versions of the Anubis cipher; the original implementation uses a pseudo- random S-box.
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In mathematics, a vexillary permutation is a permutation μ of the positive integers containing no subpermutation isomorphic to the permutation (2143); in other words, there do not exist four numbers i < j < k < l with μ(j) < μ(i) < μ(l) < μ(k). They were introduced by . The word "vexillary" means flag-like, and comes from the fact that vexillary permutations are related to flags of modules. showed that vexillary involutions are enumerated by Motzkin numbers.
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Therefore, the telephone numbers also count involutions. The problem of counting involutions was the original combinatorial enumeration problem studied by Rothe in 1800 and these numbers have also been called involution numbers... In graph theory, a subset of the edges of a graph that touches each vertex at most once is called a matching. The number of different matchings of a given graph is important in chemical graph theory, where the graphs model molecules and the number of matchings is known as the Hosoya index. The largest possible Hosoya index of an -vertex graph is given by the complete graphs, for which any pattern of pairwise connections is possible; thus, the Hosoya index of a complete graph on vertices is the same as the th telephone number.. A standard Young tableau A Ferrers diagram is a geometric shape formed by a collection of squares in the plane, grouped into a polyomino with a horizontal top edge, a vertical left edge, and a single monotonic chain of horizontal and vertical bottom and right edges.
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Duality of polytopes and order-theoretic duality are both involutions: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure. A planar graph in blue, and its dual graph in red. From any three- dimensional polyhedron, one can form a planar graph, the graph of its vertices and edges.
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There are now several short proofs of the rule, such as , and using Bender-Knuth involutions. used the Littelmann path model to generalize the Littlewood–Richardson rule to other semisimple Lie groups. The Littlewood–Richardson rule is notorious for the number of errors that appeared prior to its complete, published proof. Several published attempts to prove it are incomplete, and it is particularly difficult to avoid errors when doing hand calculations with it: even the original example in contains an error.
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The subgroup fixing one of the 759 (= 3·11·23) octads of the Golay code or Steiner system is the octad group 24:A8, order 322560, with orbits of size 8 and 16. The linear group GL(4,2) has an exceptional isomorphism to the alternating group A8. The pointwise stabilizer O of an octad is an abelian group of order 16, exponent 2, each of whose involutions moves all 16 points outside the octad. The stabilizer of the octad is a split extension of O by A8.
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Group-theoretically, the permutation representation of a regular map M is a transitive permutation group C, on a set \Omega of flags, generated by three fixed-point free involutions r0, r1, r2 satisfying (r0r2)2= I. In this definition the faces are the orbits of F = 0, r1>, edges are the orbits of E = 0, r2>, and vertices are the orbits of V = 1, r2>. More abstractly, the automorphism group of any regular map is the non-degenerate, homomorphic image of a <2,m,n>-triangle group.
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Then as a variable line lies on the point, find the locus of the midpoint of the segment determined by the planes. Young's solution starts with a line p through the point and parallel to the intersection of the planes. She identified the locus as a hyperbolic cylinder through use of a third parallel midway between the others that is the projective harmonic conjugate of a line at infinity.AMM 31(7): 356 In a triangle ABC the feet of the altitudes and midpoints of the sides are used to define three involutions.
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The problem was to show that the double points of these involutions are three pairs of opposite vertices of a complete quadrilateral. Young's solution used the radical axis of the circumcircle and nine-point circle of the triangle.AMM 37(7): 383 Young proposed construction of a strophoid: Form triangle AOB from a fixed point A and a variable B on circle centered at O. Then the locus of the orthocenter of AOB is a strophoid.AMM 38(3): 170 Another problem required the concurrence of three lines determined by a triangle's altitudes and angle bisectors.
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Bonahon received in 1972 his baccalauréat, and was accepted in 1974 into the École Normale Supérieure. He received in 1975 his maîtrise in mathematics from the University of Paris VII, and in 1979 his doctorate from the University of Paris XI under Laurence Siebenmann with thesis Involutions et fibrés de Seifert dans les variétés de dimension 3. As a postdoc he was for the academic year 1979/80 a Procter Fellow at Princeton University. In 1980 he became an attaché de recherche and in 1983 a chargé de recherche of the CNRS.
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Waldecker is the author of the book Isolated Involutions in Finite Groups (Memoirs of the American Mathematical Society, 2013), developed from her doctoral dissertation. With Lasse Rempe-Gillen, she is the coauthor of Primzahltests für Einsteiger: Zahlentheorie, Algorithmik, Kryptographie (Vieweg+Teubner, 2009; 2nd ed., Springer, 2016), a book on primality tests that was translated into English as Primality Testing for Beginners (Student Mathematical Library 70, American Mathematical Society, 2014). She became a coauthor to the 2012 textbook Elementare Algebra und Zahlentheorie of , in its second edition (Mathematik Kompakt, Springer, 2019).
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A standard Young tableau is formed by placing the numbers from 1 to into these squares in such a way that the numbers increase from left to right and from top to bottom throughout the tableau. According to the Robinson–Schensted correspondence, permutations correspond one-for-one with ordered pairs of standard Young tableaux. Inverting a permutation corresponds to swapping the two tableaux, and so the self-inverse permutations correspond to single tableaux, paired with themselves.A direct bijection between involutions and tableaux, inspired by the recurrence relation for the telephone numbers, is given by .
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The work of Elena Santiago has been studied in the context of the Congress of Contemporary Literature and in that of women novelists in the literary panorama of the 20th century. It was also the subject of the doctoral thesis defended by Dr. Muriel Taján, with the title Mythe personnel et écriture dans l'oeovre d'Elena Santiago, Évolutions et involutions d'une quête de l'absante (2009). Taján is the author of the prologue to the novel Nunca el ovido that Santiago published in 2015, after a period of six years of absence.
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Along much of the cliffed coast between Brighton and Newhaven (and beyond to Eastbourne), frost involution structures can be seen in the upper meter of the chalk cliff. These involutions appear as repeated 'U' (festoon) shaped structures and date from the Devensisn (last glacial period). Frost heaving broke up the sub-surface sedimentary layers of chalk turning small slabs into a more and more upright orientation towards the surface, to be covered by present day soil. The centre of the 'U' structure is a fill of finer chalk and sands giving a more orange / yellow appearance.
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Don Zagier used these observations to give a one-sentence proof of Fermat's theorem on sums of two squares, by describing two involutions on the same set of triples of integers, one of which can easily be shown to have only one fixed point and the other of which has a fixed point for each representation of a given prime (congruent to 1 mod 4) as a sum of two squares. Since the first involution has an odd number of fixed points, so does the second, and therefore there always exists a representation of the desired form..
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Minnaert University building, Utrecht, 2007 Corrugated panels at the Shipping and Transport College in Rotterdam. MAS (Museum aan de Stroom), Antwerp, completed 2011 The work of Neutelings Riedijk Architects has been characterized as having a sculptural, often anthropomorphic quality and a playfulness of form while following a clear rationality in programming and context.Aaron Betsky and Adam Eeuwens, False Flat: Why Dutch Design Is so Good, (New York: Phaidon, 2004) Their use of familiar forms and materials grounds the strangeness and baroque involutions that give the works a distinct identity and power.Aaron Betsky, ‘Plain weirdness: The Architecture of Neutelings Riedijk’ in Volume.
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Ly is one of the 26 sporadic groups and was discovered by Richard Lyons and Charles Sims in 1972-73. Lyons characterized 51765179004000000 as the unique possible order of any finite simple group where the centralizer of some involution is isomorphic to the nontrivial central extension of the alternating group A11 of degree 11 by the cyclic group C2. proved the existence of such a group and its uniqueness up to isomorphism with a combination of permutation group theory and machine calculations. When the McLaughlin sporadic group was discovered, it was noticed that a centralizer of one of its involutions was the perfect double cover of the alternating group A8.
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This suggested considering the double covers of the other alternating groups An as possible centralizers of involutions in simple groups. The cases n ≤ 7 are ruled out by the Brauer–Suzuki theorem, the case n = 8 leads to the McLaughlin group, the case n = 9 was ruled out by Zvonimir Janko, Lyons himself ruled out the case n = 10 and found the Lyons group for n = 11, while the cases n ≥ 12 were ruled out by J.G. Thompson and Ronald Solomon. The Schur multiplier and the outer automorphism group are both trivial. Since 37 and 67 are not supersingular primes, the Lyons group cannot be a subquotient of the monster group.
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The Feit–Thompson theorem showed that the classification of finite simple groups using centralizers of involutions might be possible, as every nonabelian simple group has an involution. Many of the techniques they introduced in their proof, especially the idea of local analysis, were developed further into tools used in the classification. Perhaps the most revolutionary aspect of the proof was its length: before the Feit–Thompson paper, few arguments in group theory were more than a few pages long and most could be read in a day. Once group theorists realized that such long arguments could work, a series of papers that were several hundred pages long started to appear.
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Let N be a rational prime, and define :J0(N) = J as the Jacobian variety of the modular curve :X0(N) = X. There are endomorphisms Tl of J for each prime number l not dividing N. These come from the Hecke operator, considered first as an algebraic correspondence on X, and from there as acting on divisor classes, which gives the action on J. There is also a Fricke involution w (and Atkin–Lehner involutions if N is composite). The Eisenstein ideal, in the (unital) subring of End(J) generated as a ring by the Tl, is generated as an ideal by the elements : Tl − l - 1 for all l not dividing N, and by :w + 1.
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In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non- trivial 2-subgroup of G. When G is a group of Lie type of characteristic 2 type, the width is usually the rank (the dimension of a maximal torus of the algebraic group).
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In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".
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The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions 𝕆, and the Frobenius theorem says the only real associative division algebras are ℝ, ℂ, and ℍ. In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8. It was matrix algebra that harnessed the hypercomplex systems. First, matrices contributed new hypercomplex numbers like 2 × 2 real matrices.
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Peña's philosophy of history acknowledges collective minds, which supervene on individual minds. No society can exist without a common memory and common plans of living-together and reaching common aims (which does not mean that all members of the body politic share those feelings; Peña rejects any mandatory imposition of beliefs or values). Peña does not deny the existence of historic breaks caused by social involutions and disasters (wars, foreign subjugations, natural catastrophes) but thinks that every human society finds its way to restart the ascending march. Peña maintains that future-oriented improvement is the sense of human life, both individual and collective; so much so that a fundamental right of man is the right to have a better life—as far as possible.
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It forms the epithelial lining of the whole of the digestive tract except part of the mouth and pharynx and the terminal part of the rectum (which are lined by involutions of the ectoderm). It also forms the lining cells of all the glands which open into the digestive tract, including those of the liver and pancreas; the epithelium of the auditory tube and tympanic cavity; the trachea, bronchi, and alveoli of the lungs; the bladder and part of the urethra; and the follicle lining of the thyroid gland and thymus. The endoderm forms: the pharynx, the esophagus, the stomach, the small intestine, the colon, the liver, the pancreas, the bladder, the epithelial parts of the trachea and bronchi, the lungs, the thyroid, and the parathyroid.
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The complete graph has ten matchings, corresponding to the value of the fourth telephone number. In mathematics, the telephone numbers or the involution numbers are a sequence of integers that count the ways telephone lines can be connected to each other, where each line can be connected to at most one other line. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on vertices, the number of permutations on elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated,.
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Another family of unitals, based on Ree groups was constructed by H. Lüneburg. Let Γ = R(q) be the Ree group of type 2G2 of order (q3 \+ 1)q3(q − 1) where q = 32m+1. Let P be the set of all q3 \+ 1 Sylow 3-subgroups of Γ. Γ acts doubly transitively on this set by conjugation (it will be convenient to think of these subgroups as points that Γ is acting on.) For any S and T in P, the pointwise stabilizer, ΓS,T is cyclic of order q - 1, and thus contains a unique involution, μ. Each such involution fixes exactly q + 1 points of P. Construct a block design on the points of P whose blocks are the fixed point sets of these various involutions μ.
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The term reflection is loose, and considered by some an abuse of language, with inversion preferred; however, point reflection is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identity map – which is also true of other maps called reflections. More narrowly, a reflection refers to a reflection in a hyperplane (n-1 dimensional affine subspace – a point on the line, a line in the plane, a plane in 3-space), with the hyperplane being fixed, but more broadly reflection is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension k, where 1 \leq k \leq n-1) is called the mirror. In dimension 1 these coincide, as a point is a hyperplane in the line.
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The Bender–Knuth involutions σk are defined for integers k, and act on the set of semistandard skew Young tableaux of some fixed shape μ/ν, where μ and ν are partitions. It acts by changing some of the elements k of the tableau to k + 1, and some of the entries k + 1 to k, in such a way that the numbers of elements with values k or k + 1 are exchanged. Call an entry of the tableau free if it is k or k + 1 and there is no other element with value k or k + 1 in the same column. For any i, the free entries of row i are all in consecutive columns, and consist of ai copies of k followed by bi copies of k + 1, for some ai and bi.
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If a is an idempotent of the endomorphism ring EndR(M), then the endomorphism is an R module involution of M. That is, f is an R homomorphism such that f 2 is the identity endomorphism of M. An idempotent element a of R and its associated involution f gives rise to two involutions of the module R, depending on viewing R as a left or right module. If r represents an arbitrary element of R, f can be viewed as a right R-homomorphism so that , or f can also be viewed as a left R module homomorphism , where . This process can be reversed if 2 is an invertible element of R:Rings in which 2 is not invertible are not hard to find. The element 2 is not invertible in any Boolean algebra, nor in any ring of characteristic 2.
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Rothe used this fact to show that the determinant of a matrix is the same as the determinant of the transpose: if one expands a determinant as a polynomial, each term corresponds to a permutation, and the sign of the term is determined by the parity of its number of inversions. Since each term of the determinant of the transpose corresponds to a term of the original matrix with the inverse permutation and the same number of inversions, it has the same sign, and so the two determinants are also the same.. In his 1800 work on permutations, Rothe also was the first to consider permutations that are involutions; that is, they are their own inverse, or equivalently they have symmetric Rothe diagrams. He found the recurrence relation :T(n) = T(n-1) + (n-1)T(n-2) for counting these permutations, which also counts the number of Young tableaux, and which has as its solution the telephone numbers :1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... ., pp.
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