Michael D. Weiner - cp's OEIS frontend

A372310 Number of permutations of length n avoiding the pattern 1324 and with 1 appearing before n.

1, 3, 11, 45, 198, 919, 4446, 22239, 114347, 601722, 3229614, 17632437, 97707195, 548538588, 3115293151, 17875151109, 103511938302, 604392787819, 3555410248782, 21057224371290, 125484804821226, 752020468811244, 4530163818778839, 27419805899781843, 166694596163875206

Offset: 2

Michael D. Weiner, Apr 26 2024

nonn

This sequence counts the number of permutations of size n written in one-line notation that avoid the pattern 1324 and have the 1 appearing before the n.

			For n=4, a(4)=11 is counting the permutations (in one-line notation): 1234, 1243, 1342, 1423, 1432, 2134, 2143, 2314, 3124, 3142, 3214.

Juan B. Gil, Oscar A. Lopez, and Michael D. Weiner, A positional statistic for 1324-avoiding permutations, arXiv:2311.18227 [math.CO], 2024.

Invert of shifted A000139.

f:= proc(n) f(n):= 2*(3*n)!/((2*n+1)!*(n+1)!) end:
a:= proc(n) option remember; `if`(n=1, 1,
      add(a(n-i)*f(i), i=1..n))
    end:
seq(a(n), n=2..26);  # Alois P. Heinz, Apr 26 2024

a[1] = 1; a[n_] := a[n] = 2*Sum[a[n-k]*(3*k)!/((2*k + 1)!*(k+1)!), {k, 1, n-1}]; Table[a[n], {n, 2, 30}] (* Vaclav Kotesovec, Jul 06 2024 *)

G.f.: A(x) = (x*(B(x)-2))/(3-B(x)), where B(x) is the g.f. for A000139. (See arxiv paper by Gil, Lopez, Weiner.)
G.f. satisfies 0 = x^4*(8*x-1) + x^2*(9*x-1)*(4*x-1)*A(x) + x*(6*x-1)*(9*x-2)*A(x)^2 + (27*x^2-9*x+1)*A(x)^3.
a(n) ~ 3^(3*n - 9/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n - 5)). - Vaclav Kotesovec, Jul 06 2024
D-finite with recurrence 4*(n-1)*(2*n-5)*a(n) -6*(n-2)*(23*n-63)*a(n-1) +9*(99*n^2-567*n+818)*a(n-2) +81*(-31*n^2+215*n-374)*a(n-3) +243*(3*n-13)*(3*n-14)*a(n-4)=0. - R. J. Mathar, Sep 27 2024

A298358 a(n) is the number of rooted 3-connected bicubic planar maps with 2n vertices.

Original entry on oeis.org

1, 0, 0, 1, 0, 3, 7, 15, 63, 168, 561, 1881, 6110, 21087, 72174, 250775, 883116, 3125910, 11174280, 40209852, 145590720, 530358095, 1941862860, 7144623447, 26403493545, 97971775008, 364903633215, 1363847131450, 5113975285788, 19233646581282

Offset: 1

Michael D. Weiner, Jan 17 2018

nonn

			A(x) = x + x^4 + 3*x^6 + 7*x^7 + 15*x^8 + 63*x^9 + 168*x^10 + 561*x^11 + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..501
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
W. T. Tutte, A census of planar maps, Canad. J. Math., 15(1963), 249-271.

Cf. A000257, A069728, A069729.

kmax = 30; b[0] = 1; b[n_] := 3*2^(n - 1)*CatalanNumber[n]/(n + 2);
G[x_] = Sum[b[k] x^k, {k, 0, kmax}];
A[_] = 1;
Do[A[x_] = G[x/(1 + A[x] + O[x]^k)^3] - 1 // Normal, {k, 1, kmax + 1}];
CoefficientList[A[x], x][[2 ;; -2]] (* Jean-François Alcover, Jun 19 2018 *)

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x), g=(1+4*y-y^2)/4);
  Vec(subst(g-1, 'x, serreverse(x*g^3)));
};
seq(30) \\ Gheorghe Coserea, Apr 11 2018

G.f.: A(x) = G(x/(1+A(x))^3)-1 where A(x*(G(x))^3) = G(x)-1 and G(x) = g.f. for A000257.
From Gheorghe Coserea, Apr 16 2018: (Start)
y = A(x)/x satisfies: 0 = x^6*y^7 + 6*x^5*y^6 + 15*x^4*y^5 + 4*x^3*(5 - 3*x)*y^4 + x^2*(15 - 37*x)*y^3 + x*(16*x^2 - 39*x + 6)*y^2 + (24*x^2 - 15*x + 1)*y + (9*x - 1).
A(x) = serreverse((1+x)^3*(1 + 12*x - (1-4*x)^(3/2))/(2*(4*x+3)^2)); equivalently, it can be rewritten as A(x) = serreverse((y-1)*(y^2+y-1)^3/(y^5*(3*y-2)^2)), where y = A000108(x). (End)
a(n) ~ 3 * 2^(9*n-1) / (sqrt(Pi) * 17^(5/2) * 5^(3*n - 5/2) * n^(5/2)). - Gheorghe Coserea and Vaclav Kotesovec, Apr 16 2018

A274259 Number of factor-free Dyck words with slope 7/3 and length 10n.

Original entry on oeis.org

1, 12, 570, 44689, 4223479, 441010458, 49014411306, 5685822210429, 680500195656621, 83406972284096638, 10416465145620729162, 1320749077779826216029, 169570747575202480367168, 22000830732097549119672094, 2880094468241888675318895339, 379941591968957300338548388051, 50458777676743899501139029335858

Offset: 0

Michael D. Weiner, Jun 16 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (3n,7n) that stay below the line y=7/3x and also do not contain a proper subpath of smaller size.

			a(2) = 570 since there are 570 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,14) that stay below the line y=7/3x and also do not contain a proper subpath of small size; e.g., ENNENENNNENNENNNENNN is a factor-free Dyck word but ENNENNENNEENNNNNENNN contains the factor ENNEENNNNN.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274258 (slope 5/3).

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(10*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/10) = 1 + 12*x + 570*x^2 + 44689*x^3 + ... . Equivalently, [x^n]( A(x)^(10*n) ) = binomial(10*n, 3*n) for n = 0,1,2,... . - Peter Bala, Jan 03 2020

A274258 Number of factor-free Dyck words with slope 5/3 and length 8n.

Original entry on oeis.org

1, 7, 133, 4140, 154938, 6398717, 281086555, 12882897819, 609038885805, 29481041746958, 1453894927584477, 72789271870852237, 3689808842747726368, 189006099916444293090, 9768094831949586349262, 508712466332195692590121, 26670630123516854616641671, 1406503552584980596900001922, 74559627811441047591493767590

Offset: 0

Michael D. Weiner, Jun 16 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (3n,5n) that stay below the line y=5/3x and also do not contain a proper subpath of smaller size.

			a(2) = 133 since there are 133 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,10) that stay below the line y=5/3x and also do not contain a proper subpath of small size; e.g., ENEEEENNNNENNNNN is a factor-free Dyck word but ENEENNENNNEENNNN contains the factor EENNENNN.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(8*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/8) = 1 + 7*x + 133*x^2 + 4140*x^3 + ... . Equivalently, [x^n]( A(x)^(8*n) ) = binomial(8*n, 3*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

A274257 Number of factor-free Dyck words with slope 4/3 and length 7n.

Original entry on oeis.org

1, 5, 52, 880, 17856, 399296, 9491008, 235274240, 6014201600, 157387037696, 4195621863424, 113534211297280, 3110485641494528, 86107512380129280, 2404899661362184192, 67680890349732102144, 1917436905101367443456, 54640222663002565640192, 1565130555077611323392000, 45039415225401829826232320

Offset: 0

Michael D. Weiner, Jun 16 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (3n,4n) that stay below the line y=4/3x and also do not contain a proper subpath of smaller size.

			a(2) = 52 since there are 52 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (6,8) that stay below the line y=4/3x and also do not contain a proper subpath of small size; e.g., EENEENENNENNNN is a factor-free Dyck word but ENEEENNENNENNN contains the factor EENNENN.

Daniel Birmajer, Juan B. Gil and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, Discrete Applied Mathematics, 244 (2018), 36-43.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Cf. A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274256 (slope 9/2), A274258 (slope 5/3), A274259 (slope 7/3).

m = 20; f[_] = 0;
Do[f[x_] = (1/(x+1)^4)(-(x^2 (x+1) f[x]^4) + x f[x]^6 + (x-1) x f[x]^5 - (x - 3) x (x+1)^2 f[x]^3 + x (x+1)^3 f[x]^2 + (x+1)^5) + O[x]^m, {m}];
CoefficientList[f[x], x] (* Jean-François Alcover, Sep 28 2019 *)

G.f. satisfies: 0 = x*f^6 + x*(x-1)*f^5 - x^2*(x+1)*f^4 - x*(x-3)*(x+1)^2*f^3 + x*(x+1)^3*f^2 - (x+1)^4*f + (x+1)^5. - Michael D. Weiner, Jan 14 2019

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(7*n, 3*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/7) = 1 + 5*x + 52*x^2 + 880*x^3 + .... Equivalently, [x^n]( A(x)^(7*n) ) = binomial(7*n, 3*n) for n = 0,1,2,.... - Peter Bala, Jan 01 2020

A274256 Number of factor-free Dyck words with slope 9/2 and length 11n.

Original entry on oeis.org

1, 5, 70, 1696, 49493, 1593861, 54591225, 1950653202, 71889214644, 2712628146949, 104277713515456, 4069334248174800, 160785480249706192, 6419443865094494044, 258585021917711797850, 10496205397574996367474, 428899108081734423242550, 17628723180468295514015268, 728347675604866545590505024

Offset: 0

Michael D. Weiner, Jun 16 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,9n) that stay below the line y=9/2x and also do not contain a proper subpath of smaller size.

			a(2) = 70 since there are 70 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,18) that stay below the line y=9/2x and also do not contain a proper subpath of small size; e.g., ENNENNENNNNNNENNNNNNNN is a factor-free Dyck word but ENEENENNNNNNNNNNNNNNNN contains the factor ENENNNNNNNN.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274244 (slope 7/2), A274257 (slope 4/3), A274258 (slope 5/3), A274259 (slope 7/3).

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(11*n,2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/11) = 1 + 5*x + 70*x^2 + 1696*x^3 + ... . Equivalently, [x^n]( A(x)^(11*n) ) = binomial(11*n, 2*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

A274244 Number of factor-free Dyck words with slope 7/2 and length 9n.

Original entry on oeis.org

1, 4, 34, 494, 8615, 165550, 3380923, 71999763, 1580990725, 35537491360, 813691565184, 18911247654404, 444978958424224, 10579389908116344, 253756528273411250, 6133110915783398175, 149219383150626519874, 3651756292682801022384, 89830021324956206790496, 2219945238901447637080235, 55088272581138888326634644

Offset: 0

Michael D. Weiner, Jun 15 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,7n) that stay below the line y=7/2x and also do not contain a proper subpath of smaller size.

			a(2) = 34 since there are 34 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,14) that stay below the line y=7/2x and also do not contain a proper subpath of small size; e.g., EEENNNNENNNNNNNNNN is a factor-free Dyck word but EEENNENNNNNNNNNNNN contains the factor ENNENNNNN.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274052 (slope 5/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).

Cf. A060941.

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(9*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/9) = 1 + 4*x + 34*x^2 + 494*x^3 + ... . Equivalently, [x^n]( A(x)^(9*n) ) = binomial(9*n, 2*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

A274052 Number of factor-free Dyck words with slope 5/2 and length 7n.

Original entry on oeis.org

1, 3, 13, 94, 810, 7667, 76998, 805560, 8684533, 95800850, 1076159466, 12268026894, 141565916433, 1650395185407, 19409211522550, 229984643863260, 2743097412254490, 32907239462485422, 396793477697214450, 4806417317271974580, 58460150525944945840, 713685698665966837135, 8742060290902752902340

Offset: 0

Michael D. Weiner, Jun 08 2016

nonn

a(n) is the number of lattice paths (allowing only north and east steps) starting at (0,0) and ending at (2n,5n) that stay below the line y = 5/2x and also do not contain a proper subpath of smaller size.

			a(2) = 13 since there are 13 lattice paths (allowing only north and east steps) starting at (0,0) and ending at (4,10) that stay below the line y=5/2x and also do not contain a proper subpath of small size; e.g., EEENNNENNNNNNN is a factor-free Dyck word but ENNEENENNNNNNN contains the factor ENENNNN.

Cyril Banderier and Michael Wallner, Lattice paths of slope 2/5, 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO).
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], 2016.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On factor-free Dyck words with half-integer slope, arXiv:1804.11244 [math.CO], 2018.
P. Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics, 225 (2000), 121-135.

Factor-free Dyck words: A005807 (slope 3/2), A274244 (slope 7/2), A274256 (slope 9/2), A274257 (slope 4/3), A274259 (slope 7/3).

Cf. A293946, A322631.

Conjectural o.g.f.: Let E(x) = exp( Sum_{n >= 1} binomial(7*n, 2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/7) = 1 + 3*x + 13*x^2 + 94*x^3 + ... . Equivalently, [x^n]( A(x)^(7*n) ) = binomial(7*n, 2*n) for n = 0,1,2,... . - Peter Bala, Jan 01 2020

A256752 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no hexagons.

Original entry on oeis.org

1, 3, 11, 44, 190, 859, 4015, 19248, 94117, 467575, 2353443, 11975568, 61505088, 318406927, 1659801852, 8704865907, 45898065978, 243163198928, 1293769867676, 6910165762943, 37036898772008, 199140325574519, 1073849938338566

Offset: 1

Michael D. Weiner, Apr 09 2015

nonn

			a(3)=11 because all 11 dissections of the pentagon are allowed, i.e., the null placement, 5 placements of 1 diagonal and 5 placements of two diagonals.

D. Birmajer, J. B. Gil, and M. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv:1503.05242 [math.CO], 2015.

Cf. A046736, A054515, A253194.

Rest[CoefficientList[(InverseSeries[Series[(y-2*y^2+y^5-y^6)/(1-y), {y, 0, 24}], x]-x)/x, x]]

a(n) = (1/(n+1))*Sum_{i=0..floor(n/4)} Sum_{k=i+1..n-3*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-4*i-1,k-i-1), n !== 0 (mod 4),

a(n) = ((-1)^(n/4)/(n+1))*binomial(5*n/4,n/4) + (1/(n+1))*Sum_{i=0..(n/4)-1} Sum_{k=i+1..n-3*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-4*i-1,k-i-1), n == 0 (mod 4).

A253194 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no pentagons.

Original entry on oeis.org

1, 3, 10, 39, 162, 707, 3190, 14766, 69719, 334481, 1625846, 7989908, 39631204, 198151579, 997623275, 5053274850, 25734158411, 131680565544, 676693557574, 3490897656337, 18071699948492, 93851485181749, 488815126122166

Offset: 1

Michael D. Weiner, Mar 24 2015

nonn

			a(3)=10 because the pentagon allows all but the null placement, i.e., 5 placements of 1 diagonal and 5 placements of two diagonals.

D. Birmajer, J. B. Gil, and M. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv:1503.05242 [math.CO], 2015.

Cf. A046736, A054515.

Rest[CoefficientList[(InverseSeries[Series[(y-2*y^2+y^4-y^5)/(1-y),{y,0,24}],x]-x)/x,x]]

A253194(n)=sum(i=0,(n-1)\3,sum(k=i+1,n-2*i, (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1)),if(n%3==0,(-1)^(n/3)*binomial(4*n/3,n/3)))/(n+1) \\ M. F. Hasler, Apr 07 2015

a(n) = (1/(n+1))*Sum_{i=0..floor(n/3)} Sum_{k=i+1..n-2*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1), n !== 0 (mod 3),
a(n) = ((-1)^(n/3)/(n+1))*binomial(4*n/3,n/3) + (1/(n+1))*Sum_{i=0..(n/3)-1} Sum_{k=i+1..n-2*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1), n == 0 (mod 3).
Recurrence: 275*(n-2)*(n-1)*n*(n+1)*(13962464*n^5 - 196202616*n^4 + 1069508732*n^3 - 2802358002*n^2 + 3480787751*n - 1597000860)*a(n) = 900*(n-2)*(n-1)*n*(27924928*n^6 - 406367696*n^5 + 2336399896*n^4 - 6678345644*n^3 + 9735406192*n^2 - 6526643891*n + 1424056473)*a(n-1) - 8*(n-2)*(n-1)*(1870970176*n^7 - 30033090896*n^6 + 197840216728*n^5 - 682911269612*n^4 + 1297104157966*n^3 - 1273486799084*n^2 + 486871358313*n + 21712608900)*a(n-2) - 16*(n-2)*(2569093376*n^8 - 47662201536*n^7 + 375012676176*n^6 - 1627682459628*n^5 + 4239503473896*n^4 - 6734585440155*n^3 + 6299789310412*n^2 - 3112752665481*n + 598681926090)*a(n-3) + 16*(2457393664*n^9 - 54190809728*n^8 + 518749193184*n^7 - 2816319789288*n^6 + 9492888047100*n^5 - 20388222826734*n^4 + 27407291375141*n^3 - 21462176121217*n^2 + 8117426803296*n - 745665750648)*a(n-4) - 8*(n-4)*(2*n - 7)*(4*n - 17)*(4*n - 11)*(13962464*n^5 - 126390296*n^4 + 424322908*n^3 - 631422862*n^2 + 369599799*n - 31302531)*a(n-5). - Vaclav Kotesovec, Mar 30 2015

cp's OEIS Frontend

User: Michael D. Weiner

A372310 Number of permutations of length n avoiding the pattern 1324 and with 1 appearing before n.

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A298358 a(n) is the number of rooted 3-connected bicubic planar maps with 2n vertices.

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A274259 Number of factor-free Dyck words with slope 7/3 and length 10n.

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A274258 Number of factor-free Dyck words with slope 5/3 and length 8n.

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A274257 Number of factor-free Dyck words with slope 4/3 and length 7n.

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A274256 Number of factor-free Dyck words with slope 9/2 and length 11n.

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A274244 Number of factor-free Dyck words with slope 7/2 and length 9n.

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A274052 Number of factor-free Dyck words with slope 5/2 and length 7n.

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