Masaya Tomie - cp's OEIS frontend

User: Masaya Tomie

Masaya Tomie has authored 3 sequences.

A338200 The number of similarity classes of pointed reflection spaces of residue two in an n-dimensional vector space over GF(2).

0, 0, 1, 2, 4, 6, 9, 12, 17, 21, 27, 33, 41, 48, 58, 67, 79, 90, 104, 117, 134, 149, 168, 186, 208, 228, 253, 276, 304, 330, 361, 390, 425, 457, 495, 531, 573, 612, 658, 701, 751, 798, 852, 903, 962, 1017, 1080, 1140, 1208, 1272, 1345, 1414, 1492, 1566, 1649

Offset: 1

Masaya Tomie, Oct 16 2020

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Saeid Azam, Masaya Tomie, and Yoji Yoshii, Classification of pointed reflection spaces, Osaka J. Math. (2021) Vol. 58, 563-589.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Cf. A069905.

F[n_] := If[EvenQ[n],
  n (n - 2)/8 +
   2*Sum[Length[IntegerPartitions[k, {3}]], {k, 3, n/2}] +
   Length[IntegerPartitions[(n + 2)/2, {3}]],
  2*Floor[(n - 1)/4]*Floor[(n + 1)/4] +
   2*Sum[Length[IntegerPartitions[k, {3}]], {k, 3, (n - 1)/2}] +
   Length[IntegerPartitions[(n + 1)/2, {3}]] +
   Length[IntegerPartitions[(n + 3)/2, {3}]]]
(* Second program: *)
LinearRecurrence[{1,1,0,0,-2,0,0,1,1,-1}, {0,0,1,2,4,6,9,12,17,21}, 55] (* Jean-François Alcover, Nov 13 2020 *)

concat([0,0], Vec((1 + x + x^2 - x^4 - x^5)/((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^50))) \\ Andrew Howroyd, Oct 29 2020

a(n) = (1/8)*n*(n-2) + 2*(Sum_{k=3..n/2} p(k,3)) + p((n+2)/2,3) if n is even; a(n) = 2*floor((n-1)/4)*floor((n+1)/4) + 2*(Sum_{k=3..(n-1)/2} p(k,3)) + p((n+1)/2,3) + p((n+3)/2,3) if n is odd, where p(k,3) = A069905(k) is the number of partitions of k into three parts.
From Andrew Howroyd, Oct 29 2020: (Start)
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 10.
G.f.: x^3*(1 + x + x^2 - x^4 - x^5)/((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
(End)

A321233 a(n) is the number of reflectable bases of the root system of type D_n.

Original entry on oeis.org

0, 4, 128, 4992, 241664, 14131200, 972521472, 77138231296, 6935178903552, 697359579217920, 77576992194560000, 9461629052252061696, 1255632936007234486272, 180144800985155488448512, 27786422394606966747955200, 4585649599904345055716966400, 806288164205933489807717040128

Offset: 1

Masaya Tomie, Nov 01 2018

nonn

The root systems of type D_n are only defined for n >= 4. See chapter 3 of the Humphreys reference. Sequence extended to n=1 using formula/recurrence.

J. E. Humphreys, Introduction to Lie algebras and representation theory, 2nd ed, Springer-Verlag, New York, 1972.

S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, A graph theoretical classification for reflectable bases, PRIMS, Vol 55 no 4, (2019), 689-736.

Cf. A000435, A320064.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[ (&+[ j^(j-1)*(4*x)^j/Factorial(j) :j in [1..m+3]])^k/(4*k) :k in [2..m+2]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Dec 09 2018

Rest[With[{m = 25}, CoefficientList[Series[Sum[Sum[j^(j - 1)*(4*x)^j/j!, {j, 1, m + 1}]^k/(4*k), {k, 2, m}], {x, 0, m}], x]*Range[0, m]!]] (* G. C. Greubel, Dec 09 2018 *)

a(n)={n!*polcoef(sum(m=2, n, (sum(k=1, n, k^(k-1)*(4*x)^k/k!) + O(x^(n-m+2)))^m/(4*m)), n)} \\ Andrew Howroyd, Nov 01 2018

A321233(n)=A001863(n)*(n-1)*4^(n-1) \\ M. F. Hasler, Dec 09 2018

from math import comb
def A321233(n): return 0 if n<2 else ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n<<(n-1<<1) # Chai Wah Wu, Apr 26 2023

E.g.f.: Sum_{m>=2} (1/(4*m)) (Sum_{k>=1} k^(k-1)*(4*x)^k/k!)^m.
a(n) = 2^n*A320064(n).
a(n) = (n-1)*4^(n-1)*A001863(n). - M. F. Hasler, Dec 09 2018

A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

Original entry on oeis.org

0, 1, 16, 312, 7552, 220800, 7597824, 301321216, 13545271296, 681015214080, 37879390720000, 2309968030334976, 153275504883695616, 10995166075754119168, 847974316241667686400, 69971459959477921382400, 6151490510604350965940224, 574035430519008722436489216, 56669921387839814123670994944

Offset: 1

Masaya Tomie, Oct 04 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, A graph theoretical classification for reflectable bases, PRIMS, Vol 55 no 4, (2019), 689-736.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 11.

Cf. A000435, A001863, A038057.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[(&+[j^(j-1)*(2*x)^j/Factorial(j): j in [1..m+2]])^k/(4*k): k in [2..m+1]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Dec 10 2018

nmax = 20; Rest[CoefficientList[Series[Sum[1/(4*m)*(Sum[k^(k-1)*(2*x)^k/k!, {k, 1, nmax}])^m, {m, 2, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Oct 23 2018 *)

seq(n)={Vec(serlaplace(sum(m=2, n, (sum(k=1, n, k^(k-1)*(2*x)^k/k!) + O(x^n))^m/(4*m))), -n)} \\ Andrew Howroyd, Nov 07 2018

apply( A320064(n)=A001863(n)*(n-1)<<(n-2), [1..20]) \\ Defines the function A320064. The additional apply(...) provides a check and illustration. - M. F. Hasler, Dec 09 2018

from math import comb
def A320064(n): return 0 if n<2 else ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n<Chai Wah Wu, Apr 25-26 2023

E.g.f.: Sum_{m>=2} (1/(4*m)) (Sum_{k>=1} k^(k-1)*(2*x)^k/k!)^m.
a(n) = (n-1)*2^(n-2)*A001863(n). - M. F. Hasler, Dec 09 2018
a(n) = 2^(n-2)*A000435(n). - Chai Wah Wu, Apr 25 2023

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User: Masaya Tomie

A338200 The number of similarity classes of pointed reflection spaces of residue two in an n-dimensional vector space over GF(2).

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A321233 a(n) is the number of reflectable bases of the root system of type D_n.

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A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

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Magma

Mathematica

PARI

PARI

Python

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