Andrei Asinowski - cp's OEIS frontend

A375923 Number of permutations of size n which are both two-clumped and co-two-clumped.

1, 1, 2, 6, 24, 112, 582, 3272, 19550, 122628, 800392, 5400342, 37475474, 266412680, 1934033968, 14300538652, 107471798112, 819442325086, 6329551390064, 49465665347580, 390692732060804, 3115700976866356, 25067250869113332, 203317147838575616, 1661425311693158000

Offset: 0

Andrei Asinowski, Sep 02 2024

nonn

Two-clumped permutations are (3-51-2-4, 3-51-4-2, 2-4-51-3, 4-2-51-3)-avoiding permutations. Co-two-clumped permutations are (3-15-2-4, 3-15-4-2, 2-4-15-3, 4-2-15-3)-avoiding permutations. Thus, this sequence enumerates permutations that avoid all these eight patterns.

a(n) is also the number of strong (=generic) rectangulations of size n whose strong poset is totally ordered.

Andrei Asinowski, Jean Cardinal, Stefan Felsner, and Éric Fusy, Combinatorics of rectangulations: Old and new bijections, arXiv:2402.01483 [math.CO], 2024, page 29.

Cf. A342141 (number of two-clumped permutations).
Cf. A001181 (Baxter numbers: number of (twisted-)Baxter permutations).
Cf. A348351 (number of permutations which are both twisted-Baxter and co-twisted-Baxter).

A375913 Number of strong (=generic) guillotine rectangulations with n rectangles.

Original entry on oeis.org

1, 2, 6, 24, 114, 606, 3494, 21434, 138100, 926008, 6418576, 45755516, 334117246, 2491317430, 18919957430, 146034939362, 1143606856808, 9072734766636, 72827462660824, 590852491725920, 4840436813758832, 40009072880216344, 333419662183186932, 2799687668599080296

Offset: 1

Andrei Asinowski, Sep 02 2024

nonn

Equivalently: The number of strong rectangulations with n rectangles that avoid two windmill patterns.

Andrei Asinowski, Jean Cardinal, Stefan Felsner, and Éric Fusy, Combinatorics of rectangulations: Old and new bijections, arXiv:2402.01483 [math.CO], 2024, page 37.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, Discrete Comput. Geom., 70(1):51-122, 2023. Page 99, Table 3, entry "12".

Cf. A342141 (number of strong (=generic) rectangulations).
Cf. A001181 (Baxter numbers: number of weak (=diagonal) rectangulations).
Cf. A006318 (Schröder numbers: number of weak (=diagonal) guillotine rectangulations).

A 5-variate recurrence is given in the paper Asinowski, Cardinal, Felsner, and Fusy.

A363813 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 4-5-3-1-2.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 295, 1114, 4166, 15390, 56167, 202738, 724813, 2570276, 9052494, 31702340, 110503497, 383691578, 1328039043, 4584708230, 15793983638, 54315199642, 186526735307, 639831906594, 2192754259993, 7509139583560, 25699765092254, 87913948206096

Offset: 0

Andrei Asinowski, Jun 23 2023

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-4-3 and 4-5-3-1-2.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5", "7", "8". See the Merino and Mütze reference, Table 3, entry "1234578".

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (10,-37,62,-47,16,-2).

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363811, A363812, A006012.

CoefficientList[Series[(1 - 9*x + 29*x^2 - 39*x^3 + 20*x^4 - 3*x^5)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2),{x,0,27}],x] (* Stefano Spezia, Jun 24 2023 *)

G.f.: (1 - 9*x + 29*x^2 - 39*x^3 + 20*x^4 - 3*x^5)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2).

A363811 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-3-5-4, and 4-5-3-1-2.

Original entry on oeis.org

1, 1, 2, 6, 22, 88, 362, 1488, 6034, 24024, 93830, 359824, 1357088, 5043260, 18501562, 67120024, 241169322, 859450004, 3041415520, 10699090888, 37448249502, 130518538696, 453276141238, 1569476495000, 5420784841936, 18683861676756, 64286814548706

Offset: 0

Andrei Asinowski, Jun 23 2023

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-3-5-4 and 4-5-3-1-2.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "7" and "8". See the Merino and Mütze reference, Table 3, entry "123478".

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (18,-141,630,-1767,3224,-3834,2896,-1312,320,-32).

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363812, A363813, A006012.

CoefficientList[Series[(1 - x)*(1 - 16*x + 109*x^2 - 410*x^3 + 923*x^4 - 1256*x^5 + 988*x^6 - 400*x^7 + 66*x^8 - 2*x^9)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2*(1 - 2*x)^4),{x,0,26}],x] (* Stefano Spezia, Jun 24 2023 *)

G.f.: (1 - x)*(1 - 16*x + 109*x^2 - 410*x^3 + 923*x^4 - 1256*x^5 + 988*x^6 - 400*x^7 + 66*x^8 - 2*x^9)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2*(1 - 2*x)^4).

A363812 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 3-41-2.

Original entry on oeis.org

1, 1, 2, 6, 20, 69, 243, 870, 3159, 11611, 43130, 161691, 611065, 2325739, 8907360, 34304298, 132770564, 516164832, 2014739748, 7892775473, 31022627947, 122304167437, 483513636064, 1916394053725, 7613498804405, 30313164090695

Offset: 0

Andrei Asinowski, Jun 23 2023

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-4-3 and 3-41-2.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5", "6", "7". See the Merino and Mütze reference, Table 3, entry "1234567".

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363811, A363813, A006012.

CoefficientList[Series[(1 - 3*x + 3*x^2 - Sqrt[1 - 6*x + 7*x^2 + 2*x^3 + x^4])/(2*x^2*(2 - x)),{x,0,25}],x] (* Stefano Spezia, Jun 24 2023 *)

G.f.: (1 - 3*x + 3*x^2 - sqrt(1 - 6*x + 7*x^2 + 2*x^3 + x^4))/(2*x^2*(2 - x)).

A363810 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-14-3, and 4-5-3-1-2.

Original entry on oeis.org

1, 1, 2, 6, 21, 79, 306, 1196, 4681, 18308, 71564, 279820, 1095533, 4298463, 16913428, 66769536, 264526329, 1051845461, 4197832133, 16813161765, 67571221016, 272448598737, 1101876945673, 4469106749281, 18174503562880, 74093063050412, 302753929958872

Offset: 0

Andrei Asinowski, Jun 23 2023

nonn

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-14-3 and 2-1-3-5-4.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5" and "8". See the Merino and Mütze reference, Table 3, entry "123458".

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363811, A363812, A363813, A006012.

with(gfun): seq(coeff(algeqtoseries(x^8*(-2+x)^2*F^4 - x^3*(x-1)*(-2+x)*(x^5-7*x^4+4*x^3-6*x^2+5*x-1)*F^3 - x*(x-1)*(4*x^7-22*x^6+37*x^5-42*x^4+53*x^3-35*x^2+10*x-1)*F^2 - (5*x^6-16*x^5+15*x^4-28*x^3+23*x^2-8*x+1)*(x-1)^2*F - (2*x^5-5*x^4+4*x^3-10*x^2+6*x-1)*(x-1)^2, x, F, 32, true)[1], x, n+1), n = 0..30); # Vaclav Kotesovec, Jun 24 2023

The generating function F=F(x) satisfies the equation x^8*(x - 2)^2*F^4 - x^3*(x - 1)*(x - 2)*(x^5 - 7*x^4 + 4*x^3 - 6*x^2 + 5*x - 1)*F^3 - x*(x - 1)*(4*x^7 - 22*x^6 + 37*x^5 - 42*x^4 + 53*x^3 - 35*x^2 + 10*x - 1)*F^2 - (5*x^6 - 16*x^5 + 15*x^4 - 28*x^3 + 23*x^2 - 8*x + 1)*(x - 1)^2*F - (2*x^5 - 5*x^4 + 4*x^3 - 10*x^2 + 6*x - 1)*(x - 1)^2 = 0.

A363809 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, and 2-1-3-5-4.

Original entry on oeis.org

1, 1, 2, 6, 22, 89, 378, 1647, 7286, 32574, 146866, 667088, 3050619, 14039075, 64992280, 302546718, 1415691181, 6656285609, 31436228056, 149079962872, 709680131574, 3390269807364, 16248661836019, 78109838535141, 376531187219762, 1819760165454501

Offset: 0

Andrei Asinowski, Jun 23 2023

nonn

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-3-5-4.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric pattern "7". See the Merino and Mütze reference, Table 3, entry "12347".

Andrei Asinowski and Cyril Banderier. Geometry meets generating functions: Rectangulations and permutations (2023).

Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A078482, A033321, A363810, A363811, A363812, A363813, A006012.

The generating function F=F(x) satisfies the equation x^4*(x - 2)^2*F^4 + x*(x - 2)*(4*x^3 - 7*x^2 + 6*x - 1)*F^3 + (2*x^4 - x^3 - 2*x^2 + 5*x - 1)*F^2 - (4*x^3 - 7*x^2 + 6*x - 1)*F + x^2 = 0.

A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5n (n up-steps and 4n down-steps).

Original entry on oeis.org

0, 4, 30, 250, 2245, 21221, 208129, 2098565, 21619910, 226593015, 2408424760, 25899375645, 281273231985, 3080585212120, 33986840371400, 377364606387005, 4213620859310140, 47284625533425750, 532996618440511710, 6032169040263819485, 68517222947120776290

Offset: 0

Andrei Asinowski, May 08 2020

A 4-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above the x-axis, and ends at the x-axis.

			For n = 2, the a(2) = 30 is the total number of down-steps after the last up-step in UddddUdddd, UdddUddddd, UddUdddddd, UdUddddddd, UUdddddddd (thus, 4 + 5 + 6 + 7 + 8).

Stefano Spezia, Table of n, a(n) for n = 0..900
Andrei Asinowski, Benjamin Hackl, and Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

First order differences of A002294. Cf. A062985.

Cf. A334682 (similar for 3-Dyck paths).

b:= proc(x, y) option remember; `if`(x=y, x,
     `if`(y+40, b(x-1, y-1), 0))
    end:
a:= n-> b(5*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 4*n, (5*(5*n-4)*
      (5*n-3)*(5*n-2)*(5*n-1)*n*(2869*n^3+5354*n^2+3239*n+634)*
       a(n-1))/(8*(n-1)*(4*n+3)*(2*n+1)*(4*n+5)*(n+1)*
       (2869*n^3-3253*n^2+1138*n-120)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020

a[n_] := Binomial[5*n + 6, n + 1]/(5*n + 6) - Binomial[5*n + 1, n]/(5*n + 1); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

a(n) = {binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1)} \\ Andrew Howroyd, May 08 2020

[binomial(5*(n + 1) + 1, n + 1)/(5*(n + 1) + 1) - binomial(5*n + 1, n)/(5*n + 1) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1).
a(n) = A062985(n+1, 4*n-1).
G.f.: ((1 - x)*HypergeometricPFQ([1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 5/4], 3125*x/256) - 1)/x. - Stefano Spezia, Apr 25 2023

A334680 a(n) is the total number of down-steps after the final up-step in all 2-Dyck paths of length 3n (n up-steps and 2n down-steps).

Original entry on oeis.org

0, 2, 9, 43, 218, 1155, 6324, 35511, 203412, 1184040, 6983925, 41652468, 250763464, 1521935948, 9301989144, 57203999295, 353701790376, 2197600497330, 13713291247635, 85907187607395, 540072341320050, 3406202392821375, 21545888897092560, 136655834260685220, 868897745157965328

Offset: 0

Andrei Asinowski, May 08 2020

A 2-Dyck path is a lattice path with steps U = (1, 2), d = (1, -1) that starts at (0,0), stays (weakly) above the x-axis, and ends at the x-axis.

			For n = 2, the a(2) = 9 is the total number of down-steps after the last up-step in UddUdd, UdUddd, UUdddd.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

First order differences of A001764.
The 4th column of A280759.
Cf. A062745.

alias(PS=ListTools:-PartialSums): A334680List := proc(m) local A, P, n;
A := [0,2]; P := [1,2]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A334680List(25); # Peter Luschny, Mar 26 2022

a[n_] := Binomial[3*n + 4, n + 1]/(3*n + 4) - Binomial[3*n + 1, n]/(3*n + 1); Array[a, 25, 0] (* Amiram Eldar, May 13 2020 *)

[(17 + 23*n)*binomial(3*n, n-1)/(2*n+2)/(2*n+3) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = binomial(3*(n+1) + 1, n+1)/(3*(n+1) + 1) - binomial(3*n + 1, n)/(3*n + 1).
a(n) = (17 + 23*n)*binomial(3*n, n - 1)/((2*n + 2)*(2*n + 3)).
a(n) = A062745(n+1, 2*n-1).

A334610 a(n) is the total number of down-steps after the final up-step in all 4_1-Dyck paths of length 5n (n up-steps and 4n down-steps).

Original entry on oeis.org

0, 7, 58, 505, 4650, 44677, 443238, 4507461, 46744100, 492492330, 5257084420, 56734340091, 618001356458, 6785943435960, 75033214770640, 834733624099485, 9336542892778440, 104932793226255165, 1184421713336050590, 13421053387405062290, 152613573227667516580

Offset: 0

Andrei Asinowski, May 13 2020

nonn

A 4_1-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -1, and ends at the x-axis.

			For n=1, a(1) = 7 is the total number of down-steps after the last up-step in Udddd, dUddd.

Stefano Spezia, Table of n, a(n) for n = 0..900
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002294, A334651, A334719, A334611, A334612.

a[n_] := 2 * Binomial[5*n + 7, n + 1]/(5*n + 7) - 4 * Binomial[5*n + 2, n]/(5*n + 2); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

a(n) = 2*binomial(5*(n+1)+2, n+1)/(5*(n+1)+2) - 4*binomial(5*n+2, n)/(5*n+2).

G.f.: ((1 - 2*x)*HypergeometricPFQ([2/5, 3/5, 4/5, 6/5], [3/4, 5/4, 3/2], 3125*x/256) - 1)/x. - Stefano Spezia, Apr 25 2023

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User: Andrei Asinowski

A375923 Number of permutations of size n which are both two-clumped and co-two-clumped.

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A375913 Number of strong (=generic) guillotine rectangulations with n rectangles.

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A363813 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 4-5-3-1-2.

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A363811 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-3-5-4, and 4-5-3-1-2.

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A363812 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 3-41-2.

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A363810 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-14-3, and 4-5-3-1-2.

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A363809 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, and 2-1-3-5-4.

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A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

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A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5n (n up-steps and 4n down-steps).

A334680 a(n) is the total number of down-steps after the final up-step in all 2-Dyck paths of length 3n (n up-steps and 2n down-steps).

A334610 a(n) is the total number of down-steps after the final up-step in all 4_1-Dyck paths of length 5n (n up-steps and 4n down-steps).