A322297 -id:A322297 - cp's OEIS frontend

A322298 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 9).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 20, 200, 1520, 9440, 50624, 242816, 1066496, 4361216, 16856556, 64202712, 288983580, 2160645840, 24525417780, 294825080160, 3270522114228, 32898687457422, 302696887652022, 2577419367939422, 20537905525582022, 155236628840778062

Offset: 0

Alois P. Heinz, Dec 02 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..548

Cf. A317111, A322251, A322262, A322276, A322297.

A322282 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 8).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 18, 162, 1122, 6402, 31746, 141570, 580866, 2241096, 8693256, 43232904, 362491272, 4067218584, 45304757784, 459941563224, 4236342378840, 35804034476496, 281634733757520, 2106753678778320, 15739783039815120

Offset: 0

Seiichi Manyama, Dec 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..537
David Galvin, John Engbers, and Clifford Smyth, Reciprocals of thinned exponential series, arXiv:2303.14057 [math.CO], 2023.
Ira M. Gessel, Reciprocals of exponential polynomials and permutation enumeration, arXiv:1807.09290 [math.CO], 2018.

Cf. A000142, A322251 (mod 3), A317111 (mod 4), A322276 (mod 5), A322262 (mod 6), A322297 (mod 7), A322298 (mod 9), A322283 (mod 10).

m = 28; CoefficientList[1/Normal[Exp[-x]+O[x]^8]+O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Feb 24 2019 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/sum(k=0, 7, (-x)^k/k!)))

E.g.f.: 1/(1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - x^7/7!).

A322283 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 22, 242, 2002, 13442, 77506, 397826, 1862146, 8085506, 32978946, 127758774, 482490294, 2015041314, 13111486674, 144226353414, 1835958708870, 22030803357420, 240151251989220, 2389590181956120, 21944411982069720, 187919216043135720

Offset: 0

Seiichi Manyama, Dec 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..557
David Galvin, John Engbers, and Clifford Smyth, Reciprocals of thinned exponential series, arXiv:2303.14057 [math.CO], 2023.
Ira M. Gessel, Reciprocals of exponential polynomials and permutation enumeration, arXiv:1807.09290 [math.CO], 2018.

Cf. A000142, A322251 (mod 3), A317111 (mod 4), A322276 (mod 5), A322262 (mod 6), A322297 (mod 7), A322282 (mod 8), A322298 (mod 9).

m = 31; CoefficientList[1/Normal[Exp[-x]+O[x]^10]+O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Feb 24 2019 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/sum(k=0, 9, (-x)^k/k!)))

E.g.f.: 1/(1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - x^7/7! + x^8/8! - x^9/9!).

cp's OEIS Frontend

A322298 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 9).

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A322282 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 8).

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A322283 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 10).

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Mathematica

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Formula