A268586 -id:A268586 - cp's OEIS frontend

A381299 Irregular triangular array read by rows. T(n,k) is the number of ordered set partitions of [n] with exactly k descents, n>=0, 0<=k<=binomial(n,2).

1, 1, 2, 1, 4, 4, 4, 1, 8, 12, 18, 18, 12, 6, 1, 16, 32, 60, 84, 100, 92, 76, 48, 24, 8, 1, 32, 80, 176, 300, 448, 572, 650, 658, 596, 478, 334, 206, 102, 40, 10, 1, 64, 192, 480, 944, 1632, 2476, 3428, 4300, 5008, 5372, 5356, 4936, 4220, 3316, 2392, 1556, 904, 456, 188, 60, 12, 1

Offset: 0

Geoffrey Critzer, Feb 19 2025

Let p = ({b_1},{b_2},...,{b_m}) be an ordered set partition of [n] into m blocks for some m, 1<=m<=n. A descent in p is an ordered pair (x,y) in [n]X[n] such that x is in b_i, y is in b_j, iy.
T(n,binomial(n,2)) = 1 (counts the ordered set partition ({n},{n-1},...,{2},{1})).
For n>=1, T(n,0) = 2^(n-1).
Sum_{k>=0} T(n,k)*2^k = A289545(n).
Sum_{k>=0} T(n,k)*3^k = A347841(n).
Sum_{k>=0} T(n,k)*4^k = A347842(n).
Sum_{k>=0} T(n,k)*5^k = A347843(n).
Sum_{k>=0} T(n,k)*6^k = A385408(n).
Sum_{k>=0} T(n,k)*7^k = A347844(n).
Sum_{k>=0} T(n,k)*8^k = A347845(n).
Sum_{k>=0} T(n,k)*9^k = A347846(n).
T(n,k) is the number of preferential arrangements of n labeled elements with exactly k inversions. For example, there 4 preferential rearrangements of length 3 with 1 inversion: 132, 213, 212, 131. - Kyle Celano, Aug 18 2025

			Triangle T(n,k) begins:
  1;
  1;
  2,  1;
  4,  4,  4,  1;
  8, 12, 18, 18,  12,  6,  1;
 16, 32, 60, 84, 100, 92, 76, 48, 24, 8, 1;
 ...

Alois P. Heinz, Rows n = 0..50, flattened
Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 2019-2020.
Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, Inversions in parking functions, arXiv:2508.11587 [math.CO], 2025. See Theorem 1.

Columns k=0-2 give: A011782, A001787(n-1) for n>=1, 2*A268586.
Cf. A000670 (row sums), A008302 (the cases where each block has size 1).
Cf. A125810, A161680, A240796, A289545, A347841, A347842, A347843, A347844, A347845, A347846, A385408.

b:= proc(o, u, t) option remember; expand(`if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(x^(u+j-1)*b(o-j, u+j-1, 1), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Feb 21 2025

nn = 7; B[n_] := FunctionExpand[QFactorial[n, u]]; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, u] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[1/(1 -(e[z] - 1)), {z, 0, nn}], z]] // Grid

Sum_{k=0..binomial(n,2)} k * T(n,k) = A240796(n). - Alois P. Heinz, Feb 20 2025
T(n,k) is the coefficient of q^k in n!q times the coefficient of x^n in 1/(1- x - x^2/2!_q - x^3/3!_q - ...), where n!_q= 1*(1+q)*(1+q+q^2)*...*(1+q+...+q^(n-1)). - _Ira M. Gessel, Jun 24 2025
T(n,k) = Sum_{w} 2^(asc(w)), where w runs through the set of permutations with k inversions and asc(w) is the number of ascents of w. - Kyle Celano, Aug 18 2025

A268587 Expansion of x^4(5 - 16x + 13x^2)/(1 - 2x)^4.

Original entry on oeis.org

0, 0, 0, 0, 5, 24, 85, 264, 760, 2080, 5488, 14080, 35328, 87040, 211200, 505856, 1198080, 2809856, 6533120, 15073280, 34537472, 78643200, 178061312, 401080320, 899153920, 2006974464, 4461690880, 9881780224, 21810380800, 47982837760, 105243475968

Offset: 0

Ran Pan, Feb 07 2016

a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly three east steps below y = x - 1 and no east steps above y = x+1. Details can be found in Section 4.1 in Pan and Remmel's link.

Robert Israel, Table of n, a(n) for n = 0..3270
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A268462, A268586.

Concatenation([0,0,0,0], List([3..40], n-> 2^(n-7)*(n-3)*(n+4)*(n+11)/3 )); # G. C. Greubel, May 24 2019

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( x^4*(5-16*x+13*x^2)/(1-2*x)^4 )); // G. C. Greubel, May 24 2019

F:= gfun:-rectoproc({16*a(n)-32*a(n+1)+24*a(n+2)-8*a(n+3)+a(n+4), a(0)=0, a(1)=0,a(2)=0,a(3)=0,a(4)=5,a(5)=24,a(6)=85},a(n),remember):
map(F, [$0..40]); # Robert Israel, Feb 07 2016

CoefficientList[Series[x^4 (5 -16x +13x^2)/(1-2x)^4, {x, 0, 40}], x] (* Michael De Vlieger, Feb 08 2016 *)
LinearRecurrence[{8,-24,32,-16},{0,0,0,0,5,24,85},40] (* Harvey P. Dale, Feb 22 2025 *)

concat(vector(4), Vec(x^4*(5-16*x+13*x^2)/(1-2*x)^4 + O(x^40))) \\ Colin Barker, Feb 08 2016

(x^4*(5-16*x+13*x^2)/(1-2*x)^4).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

G.f.: x^4*(5 - 16*x + 13*x^2)/(1 - 2*x)^4.
From Colin Barker, Feb 08 2016: (Start)
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 6.
a(n) = 2^(n-7)*(n-3)*(n+4)*(n+11)/3 for n > 2. (End)
E.g.f.: (33 + 60*x + 39*x^2 + (-33 + 6*x + 15*x^2 + 2*x^3)*exp(2*x))/96. - G. C. Greubel, May 24 2019

Typo in name and g.f. corrected by Georg Fischer, May 24 2019

A268598 Expansion of x^5(4 - 5x)/(1 - 2*x)^4.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 27, 120, 440, 1440, 4368, 12544, 34560, 92160, 239360, 608256, 1517568, 3727360, 9031680, 21626880, 51249152, 120324096, 280166400, 647495680, 1486356480, 3391094784, 7693402112, 17364418560, 39007027200, 87241523200, 194330492928

Offset: 0

Ran Pan, Feb 08 2016

a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly two east steps below y = x - 1 and exactly one east step above y = x+1. Details can be found in Section 4.1 in Pan and Remmel's link.

Colin Barker, Table of n, a(n) for n = 0..1000
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A268462, A268586, A268587.

concat(vector(5), Vec((4-5*x)*x^5/(1-2*x)^4 + O(x^40))) \\ Michel Marcus, Feb 08 2016

G.f.: x^5*(4 - 5*x)/(1 - 2*x)^4.
From Colin Barker, Feb 08 2016: (Start)
a(n) = 2^(n-7)*(n-4)*(n-3)*(n+3) for n>2.
a(n) = 8*a(n-1)-24*a(n-2)+32*a(n-3)-16*a(n-4) for n>3. (End)

A268599 Expansion of 2x^6(4-10x+7x^2)/(1-2*x)^5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 8, 60, 294, 1180, 4200, 13776, 42560, 125568, 357120, 985600, 2652672, 6988800, 18077696, 46018560, 115507200, 286326784, 701890560, 1703411712, 4096655360, 9771417600, 23132110848, 54384394240, 127049662464, 295069286400, 681574400000

Offset: 0

Ran Pan, Feb 08 2016

a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly two east steps below y = x - 1 and exactly two easts step above y = x + 1. Details can be found in Section 4.1 in Pan and Remmel's link.

Colin Barker, Table of n, a(n) for n = 0..1000
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A268462, A268586, A268587, A268598.

CoefficientList[Series[2 x^6 (4 - 10 x + 7 x^2)/(1 - 2 x)^5, {x, 0, 30}], x] (* Michael De Vlieger, Feb 08 2016 *)

concat(vector(6), Vec(2*x^6*(4-10*x+7*x^2)/(1-2*x)^5 + O(x^100))) \\ Colin Barker, Feb 08 2016

G.f.: 2*x^6*(4-10*x+7*x^2)/(1-2*x)^5.

a(n) = 2^(n-10)*(n-5)*(n-4)*(n^2+3*n+10) for n>3. - Colin Barker, Feb 08 2016

A268600 Expansion of 1/(2f(x)) + 1/(4 - 2g(x)), where f(x) = sqrt(1 - 4x) and g(x) = sqrt(1 + 4x).

Original entry on oeis.org

1, 2, 4, 12, 36, 132, 456, 1752, 6340, 24660, 91224, 356776, 1337896, 5250728, 19877904, 78209712, 298176516, 1175437428, 4505865144, 17789574792, 68490100536, 270739425528, 1046041377264, 4139198745552, 16039426479336, 63522770785032, 246761907761776, 977995685565072, 3807202080396240, 15098691607042000, 58884954519908896

Offset: 0

Ran Pan, Feb 08 2016

nonn

a(n) is the number of North-East lattice paths from (0,0) to (n,n) in which the total number of east steps below y = x - 1 or above y = x + 1 is even. Details can be found in Section 4.1 in Pan and Remmel's link.

			G.f. = 1 + 2*x + 4*x^2 + 12*x^3 + 36*x^4 + 132*x^5 + 456*x^6 + ... - _Michael Somos_, May 16 2022

Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Cf. A268462, A268586, A268587, A268598, A268599.

Cf. A000984, A126984.

CoefficientList[ Series[1/(2 Sqrt[1 - 4x]) + 1/(4 - 2 Sqrt[1 + 4x]), {x, 0, 25}], x] (* Robert G. Wilson v, Nov 24 2016 *)

my(x = 'x + O('x^40)); Vec(1/(2*sqrt(1-4*x)) + 1/(4 - 2*sqrt(1+4*x))) \\ Michel Marcus, Feb 11 2016

a(n) = binomial(2*n,n) - A268601(n).
G.f.: 1/(2*f(x)) + 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).
Conjecture D-finite with recurrence: -3*n*(n-1)*a(n) +8*(n-1)*(5*n-12)*a(n-1) +4*(-28*n+73)*a(n-2) -160*(2*n-5)*(2*n-7)*a(n-3) +192*(2*n-5)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jan 25 2020
a(n) = (-1)^n*A126984(n) + A268601(n). - Michael Somos, May 16 2022

A268601 Expansion of 1/(2f(x)) - 1/(4 - 2g(x)), where f(x) = sqrt(1 - 4x) and g(x) = sqrt(1 + 4x).

Original entry on oeis.org

0, 0, 2, 8, 34, 120, 468, 1680, 6530, 23960, 93532, 348656, 1366260, 5149872, 20238696, 76907808, 302903874, 1158168792, 4569270156, 17555689008, 69356428284, 267518448912, 1058057586456, 4094231982048, 16208177203764, 62887835652720, 249156625186328, 968943740083040, 3841488520364200, 14968574892499040, 59379627044952528

Offset: 0

Ran Pan, Feb 08 2016

nonn

a(n) is the number of North-East lattice paths from (0,0) to (n,n) in which the total number of east steps below y = x - 1 or above y = x + 1 is odd. Details can be found in Section 4.1 in Pan and Remmel's link.

Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Cf. A268462, A268586, A268587, A268598, A268599.

x = 'x + O('x^30); concat(vector(2), Vec(1/(2*sqrt(1-4*x)) - 1/(4 - 2*sqrt(1+4*x)))) \\ Michel Marcus, Feb 11 2016

a(n) = binomial(2*n,n) - A268600(n).
G.f.: 1/(2*f(x)) - 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).
Conjecture D-finite with recurrence: 3*n*(n-1)*a(n) -8*(n-1)*(5*n-12)*a(n-1) +4*(28*n-73)*a(n-2) +160*(2*n-5)*(2*n-7)*a(n-3) -192*(2*n-5)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jan 25 2020

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A381299 Irregular triangular array read by rows. T(n,k) is the number of ordered set partitions of [n] with exactly k descents, n>=0, 0<=k<=binomial(n,2).

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A268587 Expansion of x^4*(5 - 16*x + 13*x^2)/(1 - 2*x)^4.

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A268598 Expansion of x^5*(4 - 5*x)/(1 - 2*x)^4.

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A268599 Expansion of 2*x^6*(4-10*x+7*x^2)/(1-2*x)^5.

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A268600 Expansion of 1/(2*f(x)) + 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).

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A268601 Expansion of 1/(2*f(x)) - 1/(4 - 2*g(x)), where f(x) = sqrt(1 - 4*x) and g(x) = sqrt(1 + 4*x).

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A268587 Expansion of x^4(5 - 16x + 13x^2)/(1 - 2x)^4.

A268598 Expansion of x^5(4 - 5x)/(1 - 2*x)^4.

A268599 Expansion of 2x^6(4-10x+7x^2)/(1-2*x)^5.

A268600 Expansion of 1/(2f(x)) + 1/(4 - 2g(x)), where f(x) = sqrt(1 - 4x) and g(x) = sqrt(1 + 4x).

A268601 Expansion of 1/(2f(x)) - 1/(4 - 2g(x)), where f(x) = sqrt(1 - 4x) and g(x) = sqrt(1 + 4x).