A001181 -id:A001181 - cp's OEIS frontend

A264051 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A264078(n)) is the number of integer partitions of n having k standard Young tableaux such that no entries i and i+1 appear in the same row.

0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 0, 2, 4, 2, 1, 1, 1, 1, 1, 4, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 0, 0, 0, 1, 7, 2, 0, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 7, 3, 1, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1

Offset: 0

Christian Stump, Nov 01 2015

Row sums give A000041.

Column k=0 gives A025065(n-2) for n>=2.

			Triangle begins:
0,1,
0,1,
1,1,
1,2,
2,2,1,
2,3,0,2,
4,2,1,1,1,1,1,
4,3,1,0,0,2,2,0,1,0,1,0,0,0,1,
7,2,0,0,1,0,3,0,1,0,2,1,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,
...

Alois P. Heinz, Rows n = 0..14, flattened
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
FindStat - Combinatorial Statistic Finder, Number of standard Young tableaux of an integer partition such that no k and k+1 appear in the same row.

Cf. A000041, A001181, A237770, A264078.

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, x^h([1$n, l[]], 0),
      `if`(i<1, 0, g(n, i-1, l)+ `if`(i>n, 0,
       g(n-i, i, [i, l[]]))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2, [])):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 02 2015

h[l_, j_] := h[l, j] = If[l == {}, 1, If[l[[1]] == 0, h[ReplacePart[l, 1 -> Sequence[]], j - 1], Sum[If[i != j && l[[i]] > 0 && (i == 1 || l[[i]] > l[[i - 1]]), h[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, Length[l]} ]]]; g[n_, i_, l_] := If[n == 0 || i == 1, x^h[Join[Array[1 &, n], l], 0], If[i < 1, 0, g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Join[{i}, l]]] ]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n, {}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

Sum_{k=1..A264078(n)} k*T(n,k) = A237770(n). - Alois P. Heinz, Nov 02 2015

A347675 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of Baxter matrices of size n X k that contain the minimal number of 1's.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 6, 12, 1, 1, 20, 32, 32, 20, 1, 1, 30, 100, 22, 100, 30, 1, 1, 42, 240, 172, 172, 240, 42, 1, 1, 56, 490, 744, 92, 744, 490, 56, 1, 1, 72, 896, 2364, 956, 956, 2364, 896, 72, 1, 1, 90, 1512, 6174, 5328, 422, 5328, 6174, 1512, 90, 1

Offset: 1

N. J. A. Sloane, Sep 10 2021

			The array begins:
1,1,1,1,1,1,1,...
1,2,6,12,20,30,42,...
1,6,6,32,100,240,490,...
1,12,32,22,172,744,2364,...
1,20,100,172,92,956,5328,...
1,30,240,744,956,422,5492,...
1,42,490,2364,5328,5492,2074,...
...
The first few antidiagonals are:
1,
1,1,
1,2,1,
1,6,6,1,
1,12,6,12,1,
1,20,32,32,20,1,
1,30,100,22,100,30,1,
1,42,240,172,172,240,42,1,
1,56,490,744,92,744,490,56,1,
...

Michael S. Branicky, Table of n, a(n) for n = 1..96
Don Knuth, Baxter matrices, Preprint, Sep 05 2021.

Cf. A347672, A347676.

The main diagonal is A001181.

a(n) <= A347672(n). - Michael S. Branicky, Sep 15 2021

a(46)-a(66) from Michael S. Branicky, Sep 14 2021

A368708 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -2).

Original entry on oeis.org

1, 1, 3, 13, 69, 417, 2763, 19609, 146793, 1146833, 9278595, 77292261, 659973933, 5756169681, 51137399979, 461691066417, 4228199347281, 39216540096993, 367890444302787, 3486697883136957, 33353178454762389, 321754445379041601, 3127955713554766923, 30624486778208481993, 301790556354721667769, 2991957347531210976817

Offset: 0

Joerg Arndt, Jan 04 2024

Cf. A001181, A007724, A056939, A217800, A359363, A368709.

seq(simplify( hypergeom([-1 - n, -n, 1 - n], [2, 3], -2) ), n = 0..25); # Peter Bala, Sep 09 2024

Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -2], {n, 0, 30}] (* Vaclav Kotesovec, Jan 04 2024 *)
a[0] := 1; a[n_] := 2^(n + 1)/(n*(n + 1)^2)*Sum[(1/2)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1], {k, 1, n}]; Table[a[n], {n, 0, 25}] (* Detlef Meya, May 28 2024 *)

def A368708(n):
    if n == 0: return 1
    return sum(2**k * v for k, v in enumerate(A359363Row(n))) // 2
print([A368708(n) for n in range(26)]) # Peter Luschny, Jan 04 2024

def A368708(n): return PolyA359363(n, 2) // 2 if n > 0 else 1
print([A368708(n) for n in range(23)])  # Peter Luschny, Jan 04 2024

a(n) = (1/2)*B(n, 2) where B(n, x) are the Baxter polynomials with coefficients A359363, for n > 0. - Peter Luschny, Jan 04 2024
a(n) ~ 3^(n + 7/6) * (2^(2/3) + 2^(1/3) + 1)^(n + 5/3) / (2^(4/3) * Pi * n^4). - Vaclav Kotesovec, Jan 04 2024
a(0) = 1, a(n) = 2^(n + 1)/(n*(n + 1)^2)*Sum_{k=1..n} (1/2)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1). - Detlef Meya, May 29 2024
From Peter Bala, Sep 09 2024: (Start)
a(n+1) = Sum_{k = 0..n} A056939(n, k)*2^k.
P-recursive: (n+1)*(n+3)*(n+2)*(3*n-2)*a(n) = 3*(9*n^3+3*n^2-4*n+4)*(n+1)*a(n-1) + 3*(n-2)*(3*n-1)*(9*n^2-3*n-10)*a(n-2) + 27*(3*n+1)*(n-3)*(n-2)^2*a(n-3) = 0 with a(0) = 1, a(1) = 1 and a(2) = 3. (End)

A377922 Number of polyhedral orientations with n inner vertices.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 4, 15, 39, 122, 375, 1212, 3980, 13413, 45966, 160295, 566985, 2032110, 7368334, 27000759, 99891654, 372788553, 1402339329, 5313959388, 20272563265, 77822173758, 300473023929, 1166377045666, 4550358858954, 17835405078273, 70213720795466, 277552814035683

Offset: 0

N. J. A. Sloane, Dec 13 2024

nonn

a(n) is also the number of combinatorial types of corner polyhedra with n+3 flats, and the number of weak equivalence classes of tricolored contact-systems with n+3 curves. - Éric Fusy, Dec 14 2024

Éric Fusy, E. Narmanli, and G. Schaeffer, Enumeration of corner polyhedra and 3-connected Schnyder labelings, arXiv preprint arXiv:2202.09172 [math.CO], 2022-2023. See p_n.

Cf. A001181, A377920, A377921, A377923.

a(13) onwards from Éric Fusy, Dec 14 2024

A001185 Number of nontrivial Baxter permutations of length 2n-1.

Original entry on oeis.org

0, 1, 1, 7, 21, 112, 456, 2603, 13203

Offset: 1

N. J. A. Sloane

W. M. Boyce, Generation of a class of permutations associated with commuting functions, Math. Algorithms, 2 (1967), 19-26.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. M. Boyce, Generation of a class of permutations associated with commuting functions, 1967; annotated and corrected scanned copy.
W. M. Boyce, Correction to page 25 (Part 1)
W. M. Boyce, Correction to Page 25 (Part 2)
W. M. Boyce, Baxter Permutations and Functional Composition, Houston Journal of Mathematics, Volume 7, No. 2, 1981

Cf. A001181, A001183.

A342967 a(n) = 1 + Sum_{j=1..n} Product_{k=0..j-1} binomial(2n-1,n+k) / binomial(2n-1,k).

Original entry on oeis.org

1, 2, 5, 22, 177, 2606, 70226, 3457742, 311348897, 51177188350, 15377065068510, 8430169458379450, 8446194335222422950, 15435904380166258833482, 51546769958534244310727102, 313937270864810066000897492222, 3493348088919874482660174997662017

Offset: 0

Seiichi Manyama, Apr 01 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..87

Row sums of A342972.

Cf. A001181, A005362, A005363, A005364, A005365, A005366, A116925.

a[n_] := 1 + Sum[Product[Binomial[2*n - 1, n + k]/Binomial[2*n - 1, k], {k, 0, j - 1}], {j, 1, n}]; Array[a, 17, 0] (* Amiram Eldar, Apr 01 2021 *)
Table[1 + BarnesG[2*n + 1] * Sum[BarnesG[j + 1]*BarnesG[n - j + 1] / (BarnesG[n + j + 1]*BarnesG[2*n - j + 1]), {j, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 02 2021 *)

a(n) = 1+sum(j=1, n, prod(k=0, j-1, binomial(2*n-1, n+k)/binomial(2*n-1, k)));

a(n) = sum(j=0, n, prod(k=0, n-1, binomial(n+k, j)/binomial(j+k, j)));

a(n) = Sum_{j=0..n} Product_{k=0..n-1} binomial(n+k,j)/binomial(j+k,j).

a(n) ~ c * exp(1/12) * 2^(4*n^2 - 1/12) / (A * n^(1/12) * 3^(9*n^2/4 - 1/6)), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd, and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 02 2021

A348351 Number of permutations of [n] avoiding the patterns 2-41-3, 3-14-2, 2-14-3, and 3-41-2.

Original entry on oeis.org

1, 1, 2, 6, 20, 72, 274, 1088, 4470, 18884, 81652, 360054, 1614618, 7346688, 33856008, 157777908, 742637416

Offset: 0

Manfred Scheucher, Oct 20 2021

Also the number of one-sided rectangulations.

L. J. Leifheit, Combinatorial Properties of Rectangulations, Master's thesis, Technische Universität Berlin, 2021.

A. Asinowski, G. Barequet, M. Bousquet-Mélou, T. Mansour, and R. Y. Pinter, Orders induced by segments in floorplan partitions and (2-14-3,3-41-2)-avoiding permutations, Electronic Journal of Combinatorics 20(2), 2013.
Andrei Asinowski, Jean Cardinal, Stefan Felsner, and Éric Fusy, Combinatorics of rectangulations: Old and new bijections, arXiv:2402.01483 [math.CO], 2023. See p. 30.
D. Eppstein, E. Mumford, B. Speckmann, and K. Verbeek, Area-Universal Rectangular Layouts, arXiv:0901.3924 [cs.CG], 2009.
A. Merino and T. Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Manfred Scheucher, L.J. Leifheit's python program for enumeration.
The Combinatorial Object Server, Rectangulation Generator.

Cf. A001181, A006318, A078482.

Name corrected by Manfred Scheucher, May 24 2023

a(0)=1 prepended by Alois P. Heinz, Feb 05 2024

A356111 The number of 1+1+1-free ordered posets of [n].

Original entry on oeis.org

1, 1, 2, 6, 23, 102, 497, 2586, 14127, 80146, 468688, 2810163, 17206549, 107261051, 679096359, 4358360362, 28309516828, 185862601727, 1232042778231, 8238155634738, 55521191613041, 376888928783870, 2575334987109807, 17704834935517727, 122401523831513816

Offset: 0

David Bevan, Jul 27 2022

nonn

A partial order R on [n] is ordered if xRy implies x < y; i.e., the natural order (<) is a linear extension of R. 1+1+1-free posets are those with width (longest antichain) at most 2.

			The six 1+1+1-free ordered posets of [3] are those with covering relations {(1,2)}, {(1,3)}, {(2,3)}, {(1,2), (1,3)}, {(1,2), (2,3)} and {(1,3), (2,3)}.

See A006455 for the number of all ordered posets on [n], and A135922 for the number of ordered posets on [n] with height at most two.

Cf. A001181.

Conjectured g.f.: 2 - 2*x/(B(x)-1+x), where B(x) is the o.g.f. for A001181. - Michael D. Weiner, Oct 04 2024

A368709 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], +2).

Original entry on oeis.org

1, 1, -1, -3, 13, 17, -241, 121, 5081, -13327, -106705, 609589, 1850661, -23392159, -6796193, 811545073, -1688514383, -25224774367, 123764707231, 650087614573, -6385330335427, -9591188592399, 279171512779759, -318526766092183, -10665705513959287, 40625771132796817

Offset: 0

Joerg Arndt, Jan 04 2024

sign

Cf. A368708, A001181, A007724, A217800, A359363.

Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, 2], {n, 0, 30}] (* Vaclav Kotesovec, Jan 04 2024 *)
a[0] := 1; a[n_] := (-1)^n*2^(n + 1)/(n*(n + 1)^2)*Sum[(-1/2)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1], {k , 1, n}]; Table[a[n], {n, 0, 25}] (* Detlef Meya, May 29 2024 *)

def A368709(n):
    if n == 0: return 1
    return sum((-2)**k * v for k, v in enumerate(A359363Row(n))) // (-2)
print([A368709(n) for n in range(26)]) # Peter Luschny, Jan 04 2024

def A368709(n): return PolyA359363(n, -2) // (-2) if n > 0 else 1
print([A368709(n) for n in range(0, 26)])  # Peter Luschny, Jan 04 2024

a(n) = (-1/2)*B(n, -2) where B(n, x) are the Baxter polynomials with coefficients A359363, for n > 0. - Peter Luschny, Jan 04 2024

a(0) = 1, a(n) = (-1)^n*2^(n + 1)/(n*(n + 1)^2)*Sum_{k=1..n} (-1/2)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1). - Detlef Meya, May 29 2024

A214358 Number of (2-14-3, 3-41-2)-avoiding permutations of size n.

Original entry on oeis.org

1, 1, 2, 6, 22, 88, 374, 1668, 7744, 37182, 183666, 929480, 4803018, 25274088, 135132886, 732779504, 4023875702, 22346542912, 125368768090, 709852110576, 4053103780006, 23320440656376, 135126739754922, 788061492048436, 4623591001082002, 27277772831911348

Offset: 0

Mireille Bousquet-Mélou, Jul 13 2012

a(n) is also the number of permutations obtained by retaining only the even entries in a complete Baxter permutation of length 2n+1.

			For n=4, the two permutations not in this class are 2143 and 3412.

W. M. Boyce, Generation of a class of permutations associated with commuting functions, Math. Algorithms, 2 (1967), 19-26.

Alois P. Heinz, Table of n, a(n) for n = 0..400
A. Asinowski, G. Barequet, M. Bousquet-Mélou, T. Mansour, R. Pinter, Orders induced by segments in floorplans and (2-14-3,3-41-2)-avoiding permutations, Electronic Journal of Combinatorics, 20:2 (2013), Paper P35; also arXiv preprint arXiv:1011.1889 [math.CO], 2010-2012.
W. M. Boyce, Baxter Permutations and Functional Composition, Houston Journal of Mathematics, Volume 7, No. 2, 1981
F. R. K. Chung, R. L. Graham, V. E. Hoggatt Jr. and M. Kleiman, The number of Baxter permutations J. Combin. Theory Ser. A 24 (1978), no. 3, 382-394.
CombOS - Combinatorial Object Server, Generate block-aligned rectangulations
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Arturo Merino, Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Jannik Silvanus, Improved Cardinality Bounds for Rectangle Packing Representations, Doctoral Dissertation, University of Bonn (Rheinische Friedrich Wilhelms Universität, Germany 2019).

Cf. A001181.

a:= proc(n) option remember;
      `if`(n<6, [1, 1, 2, 6, 22, 88][n+1], ((8*n^3+240-8*n-48*n^2)*a(n-6)+
      (80*n-576-32*n^3+144*n^2)*a(n-5)+ (462+41*n^3-158*n^2-129*n)*a(n-4)+
      (-11*n^3-138+104*n^2+85*n)*a(n-3)+ (-14*n^3-80*n^2-92*n-30)*a(n-2)+
      (9*n^3+46*n^2+81*n+48)*a(n-1)) / ((n+4)*(n+3)*(n+1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 13 2012

a[n_] := a[n] = If[n<6, {1, 1, 2, 6, 22, 88}[[n+1]], ((8*n^3 + 240 - 8*n - 48*n^2)* a[n-6] + (80*n - 576 - 32*n^3 + 144*n^2)*a[n-5] + (462 + 41*n^3 - 158*n^2 - 129*n) *a[n-4] + (-11*n^3 - 138 + 104*n^2 + 85*n)*a[n-3] + (-14*n^3 - 80*n^2 - 92*n - 30 )*a[n-2] + (9*n^3 + 46*n^2 + 81*n + 48)*a[n-1]) / ((n+4)*(n+3)*(n+1))]; Table[ a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

The coefficients are P-recursive:
a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 6, a(4) = 22, a(5) = 88 and
(-192-280*k-96*k^2-8*k^3)*a(k) +(1824+32*k^3+432*k^2+1648*k)*a(k+1)+ (-2856-41*k^3-580*k^2-2403*k)*a(k+2) +(-1740+11*k^3+94*k^2-145*k)*a(k+3)+ (6486+14*k^3+332*k^2+2564*k)*a(k+4) +(-4134-9*k^3-208*k^2-1605*k)*a(k+5)+(630+k^3+26*k^2+223*k)*a(k+6) = 0.
Equivalently, the GF is D-finite with recurrence:
12*(t-1)*(2*t-1)^3 +(104*t-338*t^2+512*t^3 -294*t^4-110*t^5 +192*t^6-48*t^7-12) * A(t) -2*t*(t-1)*(40*t^6-128*t^5+89*t^4+53*t^3-88*t^2+35*t-4) * (d/dt)A(t) -t^2*(2*t-1)*(8*t^2-8*t+1) * (t^2-t-1)*(t-1)^2 * (d^2/dt^2)A(t) = 0.
a(n) ~ 512*(3*sqrt(2)-4) * (4+2*sqrt(2))^n/(Pi*sqrt(3)*n^4). - Vaclav Kotesovec, Aug 15 2013

cp's OEIS Frontend

A264051 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A264078(n)) is the number of integer partitions of n having k standard Young tableaux such that no entries i and i+1 appear in the same row.

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A347675 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of Baxter matrices of size n X k that contain the minimal number of 1's.

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A368708 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -2).

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A377922 Number of polyhedral orientations with n inner vertices.

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A001185 Number of nontrivial Baxter permutations of length 2n-1.

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A342967 a(n) = 1 + Sum_{j=1..n} Product_{k=0..j-1} binomial(2*n-1,n+k) / binomial(2*n-1,k).

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A348351 Number of permutations of [n] avoiding the patterns 2-41-3, 3-14-2, 2-14-3, and 3-41-2.

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A356111 The number of 1+1+1-free ordered posets of [n].

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A368709 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], +2).

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A342967 a(n) = 1 + Sum_{j=1..n} Product_{k=0..j-1} binomial(2n-1,n+k) / binomial(2n-1,k).