A111111 -id:A111111 - cp's OEIS frontend

A280780 Numerators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).

1, -4, 2, -40, -182, -7624, -202652, -14115088, -30800534, -16435427656, -1051314228316, -22675483971248, -6980651581556876, -283099764343781072, -163910651754113166328, -43009695328217994139936, -793529010007812171331166, -20144221762701827321778088, -274475989492312981198559876

Offset: 0

N. J. A. Sloane, Jan 19 2017

			Coefficients are 1, -4, 2, -40/3, -182/3, -7624/15, -202652/45, -14115088/315, -30800534/63, -16435427656/2835, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..100
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280778, A280779, A280781.

seq(N) = {
  my(f = serreverse(x*Ser(vector(N, n, n!))));
  Vec(x* f'/f * exp(2 + (f-x)/(x*f)));
};
apply(numerator, seq(20))  \\ Gheorghe Coserea, Jan 22 2017

A111111(n) ~ n!*exp(-2)*(1 - 4/n + 2/(n*(n-1)) - (40/3)/(n*(n-1)*(n-2)) - ...). - Gheorghe Coserea, Jan 23 2017

More terms from Gheorghe Coserea, Jan 22 2017

A280781 Denominators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 45, 315, 63, 2835, 14175, 22275, 467775, 1216215, 42567525, 638512875, 638512875, 834978375, 558242685, 1856156927625, 713906510625, 17717861581875, 2143861251406875, 9861761756471625, 147926426347074375, 75472666503609375, 48076088562799171875

Offset: 0

N. J. A. Sloane, Jan 19 2017

Has the same start as A046983 but is a different sequence.

			Coefficients are 1, -4, 2, -40/3, -182/3, -7624/15, -202652/45, -14115088/315, -30800534/63, -16435427656/2835, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..100
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280778, A280779, A280780.

Cf. also A046983.

seq(N) = {
  my(f = serreverse(x*Ser(vector(N, n, n!))));
  Vec(x* f'/f * exp(2 + (f-x)/(x*f)));
};
apply(denominator, seq(28))  \\ Gheorghe Coserea, Jan 22 2017

More terms from Gheorghe Coserea, Jan 22 2017

A280777 Denominators of coefficients in asymptotic expansion of C_n (number of connected chord diagrams, A000699).

Original entry on oeis.org

1, 2, 8, 16, 128, 1280, 3072, 215040, 3440640, 2293760, 137625600, 201850880, 72666316800, 1889324236800, 52901078630400, 176336928768000, 135426761293824, 191854578499584000, 1593868805996544000, 787371190162292736000, 31494847606491709440000, 29395191099392262144000

Offset: 0

N. J. A. Sloane, Jan 19 2017

			Coefficients are 1, -5/2, -43/8, -579/16, -44477/128, -5326191/1280, -180306541/3072, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..101
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280776, A280778, A280779, A280780, A280781.

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
seq(N) = my(C = 'x*Ser(A000699_seq(N))); Vec(x*exp(1-(2*C+C^2)/(2*x))/C);
apply(denominator, seq(22)) \\ Gheorghe Coserea, Jan 22 2017

More terms from Gheorghe Coserea, Jan 22 2017

A280778 Numerators of coefficients in asymptotic expansion of M_n (number of monolithic chord diagrams, A280775).

Original entry on oeis.org

1, -4, -6, -154, -1610, -34588, -4666292, -553625626, -1158735422, -388434091184, -31268175015478, -2796356409576766, -4624948938397276052, -1691272281281652408568, -2154089954877183990112, -170222948041126582837968646, -5761785676811885455064909606, -55629298859254851627617870836

Offset: 0

N. J. A. Sloane, Jan 19 2017

			Coefficients are 1, -4, -6, -154/3, -1610/3, -34588/5, -4666292/45, -553625626/315, -1158735422/35, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..101
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280779, A280780, A280781.

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
seq(N) = {
  my(M = subst(x*Ser(A000699_seq(N)), x, x/(1-x)^2));
  Vec(x/(1-x)*exp(1-x/2-(1-x)^2/(2*x)*(2*M + M^2))/M);
};
apply(numerator, seq(18))  \\ Gheorghe Coserea, Jan 22 2017

More terms from Gheorghe Coserea, Jan 22 2017

A280779 Denominators of coefficients in asymptotic expansion of M_n (number of monolithic chord diagrams, A280775).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 45, 315, 35, 567, 2025, 7425, 467775, 6081075, 257985, 638512875, 638512875, 172297125, 13956067125, 74246277105, 3093594879375, 14992036723125, 2143861251406875, 16436269594119375, 4226469324202125, 48028060502296875, 593531957565421875, 56437147443285984375

Offset: 0

N. J. A. Sloane, Jan 19 2017

This has the same start as two other sequences, A241591 and A248592, but appears to be different from both.

			Coefficients are 1, -4,-6, -154/3, -1610/3, -34588/5, -4666292/45, -553625626/315, -1158735422/35, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..101
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280778, A280780, A280781.

Cf. also A241591, A248592.

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
seq(N) = {
  my(M = subst(x*Ser(A000699_seq(N)), x, x/(1-x)^2));
  Vec(x/(1-x)*exp(1-x/2-(1-x)^2/(2*x)*(2*M + M^2))/M);
};
apply(numerator, seq(18))  \\ Gheorghe Coserea, Jan 22 2017

A280776 Numerators of coefficients in asymptotic expansion of C_n (number of connected chord diagrams, A000699).

Original entry on oeis.org

1, -5, -43, -579, -44477, -5326191, -180306541, -203331297947, -58726239094693, -781618285277957, -1025587838964854273, -35763822710356866613, -330773478104531041960421, -237504847171108896327033959, -196526060612842999084524774697, -20633624138373135772483762873819

Offset: 0

N. J. A. Sloane, Jan 19 2017

			Coefficients are 1, -5/2, -43/8, -579/16, -44477/128, -5326191/1280, -180306541/3072, ...

Gheorghe Coserea, Table of n, a(n) for n = 0..101
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.

Cf. A000699, A280775, A111111, A280777, A280778, A280779, A280780, A280781.

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
seq(N) = my(C = 'x*Ser(A000699_seq(N))); Vec(x*exp(1-(2*C+C^2)/(2*x))/C);
apply(numerator, seq(16))  \\ Gheorghe Coserea, Jan 22 2017

More terms from Gheorghe Coserea, Jan 22 2017

A244522 Number of simple involutions of length n.

Original entry on oeis.org

1, 1, 2, 0, 0, 2, 4, 10, 36, 126, 454, 1744, 6892, 28486, 120962, 531660, 2398248, 11127674, 52888998, 257570104, 1282439216

Offset: 0

Joerg Arndt and Jay Pantone, Jul 21 2014

			The a(7) = 10 simple involutions are:
01:  [ 5 1 4 6 2 0 3 ]    (0, 5) (1) (2, 4) (3, 6)
02:  [ 4 1 6 3 0 5 2 ]    (0, 4) (1) (2, 6) (3) (5)
03:  [ 4 6 2 5 0 3 1 ]    (0, 4) (1, 6) (2) (3, 5)
04:  [ 5 3 6 1 4 0 2 ]    (0, 5) (1, 3) (2, 6) (4)
05:  [ 3 1 5 0 6 2 4 ]    (0, 3) (1) (2, 5) (4, 6)
06:  [ 3 5 2 0 6 1 4 ]    (0, 3) (1, 5) (2) (4, 6)
07:  [ 3 6 4 0 2 5 1 ]    (0, 3) (1, 6) (2, 4) (5)
08:  [ 2 5 0 3 6 1 4 ]    (0, 2) (1, 5) (3) (4, 6)
09:  [ 2 5 0 6 4 1 3 ]    (0, 2) (1, 5) (3, 6) (4)
10:  [ 2 4 0 6 1 5 3 ]    (0, 2) (1, 4) (3, 6) (5)

Cf. A000085 (involutions), A111111 (simple permutations).

A359856 Number of permutations of [1..n] which are indecomposable by direct and skew sums.

Original entry on oeis.org

1, 1, 0, 0, 2, 22, 202, 1854, 17866, 183806, 2029850, 24081006, 306486314, 4175102110, 60708557626, 939518187726, 15430666746826, 268214861561726, 4921023843969242, 95066628485598126, 1929291834938927210, 41042364285004263262, 913409469123533445754, 21227246586149632119438

Offset: 0

Ludovic Schwob, Jan 16 2023

nonn

			The only permutations of [1..4] which are indecomposable by direct and skew sums are 2413 and 3142.

Wikipedia, Skew and direct sums of permutations

Cf. A003319, A111111.

nmax = 20; CoefficientList[Series[2*(2 - 1/Sum[k!*x^k, {k, 0, nmax}]) - Sum[k!*x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 19 2023 *)

seq(n)={my(p=sum(k=0, n, k!*x^k, O(x*x^n))); Vec(2*(2 - 1/p) - p)} \\ Andrew Howroyd, Jan 16 2023

G.f.: 2*(2 - 1/F(x)) - F(x) where F(x) = Sum_{k>=0} k!*x^k.
G.f.: S(F(x)) - 2*F(x)^2 - F(x) + x + 1 where S(x) is the g.f. of A111111 and F(x) = Sum_{k>=1} k!*x^k.
a(n) ~ n! * (1 - 4/n - 2/n^2 - 10/n^3 - 64/n^4 - 506/n^5 - 4762/n^6 - 51824/n^7 - 638678/n^8 - 8777898/n^9 - 132990772/n^10 - ...). - Vaclav Kotesovec, Jan 19 2023

A133262 Number of two-dimensional simple permutations.

Original entry on oeis.org

1, 4, 8, 172, 5204, 222716, 12509188, 889421564, 78097622276, 8312906703868, 1056520142488580, 158263730949406716, 27626236450406776836, 5563092167972597137404, 1280742543230231763615748, 334405228960123174787678204, 98317121153947856929753989124, 32339023133437156084762282819580, 11831483864832785151824395066146820, 4789379698138059405310741712024371196

Offset: 1

Hao Zhang and Daniel Gildea (zhanghao(AT)cs.rochester.edu), Oct 15 2007

nonn

A two-dimensional permutation of n is a vector of three permutations, with the first element being the identity permutation. For example, ( (1 2 3) (1 3 2) (3 1 2) ) is a two-dimensional permutation of 3. The example is a simple two-dimensional permutation because none of the intervals of length 2 in the permutations is common among all three. On the other hand, ( (1 2 3) (1 3 2) (2 3 1) ) is not simple because the intervals covering 2 and 3 are common among all three permutations.

M. H. Albert, M. D. Atkinson and M. Klazar, The enumeration of simple permutations, Journal of Integer Sequences 6 (2003), Article 03.4.4.
Hao Zhang and Daniel Gildea, Enumeration of Factorizable Multi-Dimensional Permutations, J. Integer Sequences 10 (2007), Article 07.5.8.

Cf. A006318, A111111.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 10 2008

A198434 Number of simple symmetric permutations of degree 2n (or 2n+1).

Original entry on oeis.org

2, 10, 90, 966, 12338, 181470, 3018082, 55995486, 1146939010, 25716746430, 626755197698, 16502357651966, 466944932413442, 14133259249586174, 455715081098876418, 15596665064842012158, 564724372634695925762, 21568978799171323200510, 866674159679235417061378, 36548294282449538711357438

Offset: 2

David Wehlau, Oct 24 2011

nonn

A permutation is simple if the only intervals that are fixed are the singletons and [1..m].
A permutation p is symmetric if i+j=m+1 implies p(i)+p(j)=m+1.
For example the permutations
1234  and   12345
2413        25314
are both simple and symmetric.
Symmetric simple permutations of degree 2n+1 correspond to simple permutations in the Weyl group of type B_n.
Symmetric simple permutations of degree 2n correspond to simple permutations in the Weyl group of type C_n.
These occur in pairs so all entries in this sequence will be even.

			The simple symmetric permutations of lowest degree are 2413, 3142, 25314, 41325.

R. Dewji, I. Dimitrov, A. McCabe, M. Roth, D. Wehlau and J. Wilson,
Decomposing Inversion Sets of Permutations and Applications to Faces of the Littlewood-Richardson Cone, arXiv:1110.5880v1[math.CO]

Cf. A111111.

cp's OEIS Frontend

A280780 Numerators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).

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A280781 Denominators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).

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A280777 Denominators of coefficients in asymptotic expansion of C_n (number of connected chord diagrams, A000699).

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A280778 Numerators of coefficients in asymptotic expansion of M_n (number of monolithic chord diagrams, A280775).

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A280779 Denominators of coefficients in asymptotic expansion of M_n (number of monolithic chord diagrams, A280775).

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A280776 Numerators of coefficients in asymptotic expansion of C_n (number of connected chord diagrams, A000699).

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PARI

Extensions

A244522 Number of simple involutions of length n.

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A359856 Number of permutations of [1..n] which are indecomposable by direct and skew sums.

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