Bridget Tenner - cp's OEIS frontend

A363582 Number of admissible mesa sets among Stirling permutations of order n.

1, 2, 3, 6, 12, 22, 44, 88, 169, 338, 676, 1322, 2644, 5288, 10433, 20866, 41732, 82736, 165472, 330944, 658012, 1316024, 2632048, 5242778, 10485556, 20971112, 41822049, 83644098, 167288196, 333885702, 667771404, 1335542808, 2667053601, 5334107202, 10668214404

Offset: 1

Bridget Tenner, Jun 10 2023

nonn

			For n = 4, the a(4) = 6 admissible pinnacle sets for Stirling permutations of order 4 are {}, {2}, {3}, {4}, {2,4}, and {3,4}.

Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, "Mesas of Stirling permutations," preprint.

Alois P. Heinz, Table of n, a(n) for n = 1..3323

Cf. A289871, A359066, A359067, A328901, A328902.

a:= proc(n) option remember; `if`(n<4, n, (2*n*(2*n-3)*
      a(n-1)+27*(n-4)*(n-2)*(a(n-3)/2-a(n-4)))/(n*(2*n-3)))
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, Jun 13 2023

Let n = 3*k+r, where r is in {0,1,2}, and let C_(x,y) be the rational Catalan numbers (A328901/A328902). Then a(n) = 2^(n-1) - Sum_{i=0..k-1} 2^(3*i+r)*C_(2*(k-i)-1,k-i).

More terms from Alois P. Heinz, Jun 13 2023

A359066 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).

Original entry on oeis.org

1, 1, 5, 7, 31, 49, 209, 351, 1471, 2561, 10625, 18943, 78079, 141569, 580865, 1066495, 4361215, 8085505, 32978945, 61616127, 250806271, 471556097, 1916280833, 3621830655, 14698053631, 27902803969, 113104519169, 215530668031, 872801042431, 1668644405249, 6751535300609

Offset: 1

Bridget Tenner, Dec 15 2022

For n >= 3, this is the number of admissible pinnacle sets in the group S_n^B of signed permutations.

The even-indexed terms appear in A240721 and the odd-indexed terms appear in A178792.

			For n = 3, the a(3) = 5 admissible pinnacle sets in S_3^B are {}, {-1}, {1}, {2}, {3}.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO] (2023).

Cf. A289871, A359067, A359068, A240721, A178792.

a := n -> add(binomial(n, k)*binomial(n-1-k, iquo(n-1, 2) - k), k = 0..iquo(n-1,2)):
# Alternative:
a := n -> binomial(n-1, floor((n-1)/2))*hypergeom([-n, ceil((1-n)/2)], [1-n], -1);
seq(simplify(a(n)), n=3..31); # Peter Luschny, Jan 03 2023

Array[Sum[Binomial[#, k]*Binomial[# - 1 - k, Floor[(# - 1)/2] - k], {k, 0, Floor[(# - 1)/2]}] &, 31] (* Michael De Vlieger, Jan 03 2023 *)

a(n) = sum(k=0, (n-1)\2, binomial(n,k)*binomial(n-1-k, (n-1)\2 - k)) \\ Andrew Howroyd, Jan 02 2023

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).

a(n) = binomial(n-1, floor((n-1)/2))*hypergeom([-n, ceil((1 -n)/2)], [1 - n], -1). - Peter Luschny, Jan 03 2023

A359067 a(2n) = Sum_{k=0..n-1} binomial(2n,k) binomial(2n-1-k, n-1-k). a(2n+1) = (Sum_{k=0..n} binomial(2n+1,k) binomial(2n-k, n-k)) - binomial(2*n-1, n).

Original entry on oeis.org

0, 1, 4, 7, 28, 49, 199, 351, 1436, 2561, 10499, 18943, 77617, 141569, 579149, 1066495, 4354780, 8085505, 32954635, 61616127, 250713893, 471556097, 1915928117, 3621830655, 14696701553, 27902803969, 113099318869, 215530668031, 872780984131, 1668644405249, 6751457741849

Offset: 1

Bridget Tenner, Dec 15 2022

For n >= 3, the number of admissible pinnacle sets in the group S_n^D of even-signed permutations.

The even-indexed terms match the even-indexed terms of A359066. The odd-indexed terms differ from the odd-indexed terms of A359066 by binomial(2*n-1, n).

			For n = 3, the a(3) = 4 admissible pinnacle sets in S_3^D are {}, {1}, {2}, {3}.

Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO] (2023).

Cf. A289871, A359066, A359068, A178792, A240721.

a := n -> if irem(n - 1, 2) = 1 then binomial(n, n/2 - 1)*hypergeom([n/2 + 1, -n/2 + 1], [n/2 + 2], -1) else binomial(n + 1, n/2 + 1/2)*hypergeom([n/2 + 1/2, -n/2 + 1/2], [n/2 + 3/2], -1)/2 - binomial(n - 2, n/2 - 1/2) fi:
seq(simplify(a(n)), n = 3..31); # Peter Luschny, Jan 03 2023

a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k).
a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).
a(n) = A240721((n-2)/2) if n-1 is odd and otherwise A178792((n-1)/2) - binomial(2*n - 1, n). - Peter Luschny, Jan 03 2023

A331347 Number of permutations w in S_n that form Boolean intervals [s, w] in the Bruhat order for every simple reflection s in the support of w.

Original entry on oeis.org

1, 2, 6, 15, 37, 93, 238, 616, 1604, 4189, 10955, 28667, 75036, 196430, 514242, 1346283, 3524593, 9227481, 24157834, 63246004, 165580160, 433494457, 1134903191, 2971215095, 7778742072, 20365011098, 53316291198, 139583862471, 365435296189, 956722026069

Offset: 1

Bridget Tenner, Jan 14 2020

			a(4) = 15 because the permutations with this property in S_4 are all permutations of length < 4.

Colin Barker, Table of n, a(n) for n = 1..1000
B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).

Cf. A001519.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1-3*x+4*x^2-4*x^3+x^4)/((1-x)^2*(1-3*x+x^2)))); // Marius A. Burtea, Jan 15 2020

Join[{1},Table[Fibonacci[2n-1]+n-2,{n,2,30}]] (* or *) LinearRecurrence[ {5,-8,5,-1},{1,2,6,15,37},30] (* Harvey P. Dale, Feb 21 2020 *)

Vec(x*(1 - 3*x + 4*x^2 - 4*x^3 + x^4) / ((1 - x)^2*(1 - 3*x + x^2)) + O(x^30)) \\ Colin Barker, Jan 14 2020

a(n) = Fibonacci(2n-1) + n - 2 = A001519(n) + n - 2.
From Colin Barker, Jan 14 2020: (Start)
G.f.: x*(1 - 3*x + 4*x^2 - 4*x^3 + x^4) / ((1 - x)^2*(1 - 3*x + x^2)).
a(n) = 5*a(n-1) - 8*a(n-2) + 5*a(n-3) - a(n-4) for n>5.
(End)
E.g.f.: 1 + exp((1/2)*(3-sqrt(5))*x)*(3 + sqrt(5) + 2*exp(sqrt(5)*x))/(5 + sqrt(5)) + exp(x)*(x - 2). - Stefano Spezia, Jan 15 2020

A330503 Number of Sós permutations of {0,1,...,n}.

Original entry on oeis.org

2, 6, 16, 30, 60, 84, 144, 198, 280, 352, 504, 598, 812, 960, 1152, 1360, 1728, 1938, 2400, 2688, 3080, 3450, 4128, 4500, 5200, 5724, 6440, 7018, 8100, 8618, 9856, 10692, 11696, 12600, 13824, 14652, 16416, 17550, 18960, 20090, 22260, 23306, 25696, 27180, 28888

Offset: 1

Bridget Tenner, Dec 16 2019

			For n = 3, the a(3) = 16 Farey functions of {0,1,2,3} are {0123, 3012, 2301, 1230, 0312, 2031, 1203, 3120, 0213, 3021, 1302, 2130, 0321, 1032, 2103, 3210}.

S. Bockting-Conrad, Y. Kashina, T. K. Petersen, and B. E. Tenner, Sós permutations, arXiv:2007.01132 [math.CO], 2020.

Cf. A002088.

MapIndexed[(First[#2] + 1) #1 &, Accumulate@ Array[EulerPhi, 45]] (* Michael De Vlieger, Dec 16 2019 *)

a(n)={(n+1)*sum(k=1, n, eulerphi(k))} \\ Andrew Howroyd, Dec 20 2019

from functools import lru_cache
@lru_cache(maxsize=None)
def A330503(n):
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(2*A330503(k1)//(k1+1)-1)
        j, k1 = j2, n//j2
    return (n+1)*(n*(n-1)-c+j)//2 # Chai Wah Wu, Mar 29 2021

a(n) = (n+1) * Sum_{k=1..n} phi(k), where phi(k) is Euler's totient function.

a(n) = (n+1) * A002088(n).

More terms from Michael De Vlieger, Dec 16 2019

A320944 a(n) is the number of perimeter tiles among all rhombic tilings of the 2n-gon.

Original entry on oeis.org

1, 6, 24, 200, 3216

Offset: 2

Bridget Tenner, Oct 24 2018

Tiles and polygons all have sides of length 1, as in A006245.

			a(4) = 24 because among the eight rhombic tilings of the octagon, 24 perimeter rhombi appear.

B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.

Cf. A320945, A320946, A006245.

a(n) = 2*n*A320945(n) for n > 2.

A320946 a(n) is the number of left-perimeter tiles among all rhombic tilings of the 2n-gon.

Original entry on oeis.org

1, 2, 9, 80, 1340

Offset: 2

Bridget Tenner, Oct 24 2018

Also the number of right-perimeter tiles among all rhombic tilings of the 2n-gon. Tiles and polygons all have sides of length 1, as in A006245.

			a(4) = 9 because among the eight rhombic tilings of the octagon, 9 left-perimeter rhombi appear.

B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.

Cf. A320944, A320945, A006245.

a(n) = (n-1) * A320945(n).

A320945 a(n) is the number of top-perimeter tiles among all rhombic tilings of the 2n-gon.

Original entry on oeis.org

1, 1, 3, 20, 268

Offset: 2

Bridget Tenner, Oct 24 2018

Also the number of bottom-perimeter tiles among all rhombic tilings of the 2n-gon. Tiles and polygons all have sides of length 1, as in A006245.

			a(4) = 3 because among the eight rhombic tilings of the octagon, 3 top-perimeter rhombi appear.

B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.

Cf. A320944, A320946, A006245.

A320947 a(n) is the number of dominoes, among all domino tilings of the 2 X n rectangle, sharing a length-2 side with the boundary of the rectangle.

Original entry on oeis.org

1, 4, 8, 16, 30, 56, 102, 184, 328, 580, 1018, 1776, 3082, 5324, 9160, 15704, 26838, 45736, 77742, 131840, 223112, 376844, 635378, 1069536, 1797650, 3017236, 5057672, 8467744, 14161038, 23657240, 39482358, 65832136, 109671112, 182552404, 303629290

Offset: 1

Bridget Tenner, Oct 24 2018

a(n) is also the number of dominoes, among all domino tilings of the 2 x n rectangle, sharing a contiguous path of length at least 2 with the boundary of the rectangle.

			a(4) = 16 because among the five domino tilings of the 2 X 4 rectangle, 16 dominoes share a length 2 side with the boundary.

Colin Barker, Table of n, a(n) for n = 1..1000
B. E. Tenner, Tiling-based models of perimeter and area, arXiv:1811.00082 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045.

Rest@ CoefficientList[Series[x (1 + 2 x - x^2 - 2 x^3 - x^4)/(1 - x - x^2)^2, {x, 0, 35}], x] (* Michael De Vlieger, Nov 05 2018 *)

Vec(x*(1 + 2*x - x^2 - 2*x^3 - x^4) / (1 - x - x^2)^2 + O(x^40)) \\ Colin Barker, Nov 02 2018

a(n) = a(n-1) + a(n-2) + 2*Fibonacci(n-1) for n > 3.
From Colin Barker, Nov 02 2018: (Start)
G.f.: x*(1 + 2*x - x^2 - 2*x^3 - x^4) / (1 - x - x^2)^2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n>5.
a(n) = 2^(1-n)*(-4*sqrt(5)*((1-sqrt(5))^n - (1+sqrt(5))^n) + 5*((1-sqrt(5))^n + (1+sqrt(5))^n)*n) / 25 for n>1.
(End)

A290954 The number of permutations in S_n for which the number of reduced words is minimized with respect to the numbers of braid and commutation classes: |R(w)| = |B(w)| + |C(w)| - 1.

Original entry on oeis.org

1, 2, 6, 23, 65, 177, 506, 1551, 5041, 17049, 59131, 208469, 743491, 2675189, 9695778, 35358815, 129646177, 477640361, 1767265159, 6564122733, 24466269715, 91482566757, 343059617231, 1289904151413, 4861946406095, 18367353077397, 69533550921901, 263747951756961, 1002242216658727, 3814986502100477

Offset: 1

Bridget Tenner, Aug 15 2017

			a(3) = 6 because all six permutations in S_3 have this property.

S. Fishel, E. Milicevic, R. Patrias, and B. E. Tenner, Enumerations relating braid and commutation classes, arXiv:1708.04372 [math.CO], 2017.

Cf. A000108, A290953.

a(1) = 1, a(2) = 2, and a(n) = C(n) + (n^3 - 3n^2 + 8n - 21)/3 for n > 2, where C(n) is the n-th Catalan number.

cp's OEIS Frontend

User: Bridget Tenner

A363582 Number of admissible mesa sets among Stirling permutations of order n.

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A359066 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,k)*binomial(n-1-k,floor((n-1)/2) - k).

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A359067 a(2*n) = Sum_{k=0..n-1} binomial(2*n,k) binomial(2*n-1-k, n-1-k). a(2*n+1) = (Sum_{k=0..n} binomial(2*n+1,k) binomial(2*n-k, n-k)) - binomial(2*n-1, n).

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A331347 Number of permutations w in S_n that form Boolean intervals [s, w] in the Bruhat order for every simple reflection s in the support of w.

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Magma

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A330503 Number of Sós permutations of {0,1,...,n}.

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A320944 a(n) is the number of perimeter tiles among all rhombic tilings of the 2n-gon.

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A320946 a(n) is the number of left-perimeter tiles among all rhombic tilings of the 2n-gon.

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A359067 a(2n) = Sum_{k=0..n-1} binomial(2n,k) binomial(2n-1-k, n-1-k). a(2n+1) = (Sum_{k=0..n} binomial(2n+1,k) binomial(2n-k, n-k)) - binomial(2*n-1, n).