Derek Lim - cp's OEIS frontend

A370649 Dimension of space of equivariant linear maps from R^{n^3} to R^{n^3} under diagonal action of {-1, 1}^n.

0, 1, 32, 183, 544, 1205, 2256, 3787, 5888, 8649, 12160, 16511, 21792, 28093, 35504, 44115, 54016, 65297, 78048, 92359, 108320, 126021, 145552, 167003, 190464, 216025, 243776, 273807, 306208, 341069, 378480, 418531, 461312, 506913, 555424, 606935, 661536, 719317

Offset: 0

Derek Lim, Feb 25 2024

D. Lim, J. Robinson, S. Jegelka, and H. Maron, Expressive Sign Equivariant Networks for Spectral Geometric Learning, Conference on Neural Information Processing Systems (NeurIPS), 2023. See Table 4 at p. 24.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000567.

a:= n-> ((15*n-30)*n+16)*n:
seq(a(n), n=0..37);  # Alois P. Heinz, Jul 14 2024

def A370649(n): return n*(15*(n-1)**2+1) # Chai Wah Wu, Jul 15 2024

a(n) = (1/2^n) * Sum_{s in {-1,1}^n} (s_1 + s_2 + ... + s_n)^6 [from Proposition 7 of Lim et al.]. - Sean A. Irvine, Jul 14 2024
From Alois P. Heinz, Jul 14 2024: (Start)
a(n) = 2^(-n) * Sum_{k=0..n} (2*k-n)^6 * binomial(n,k).
G.f.: x*(61*x^2+28*x+1)/(x-1)^4.
a(n) = 15*n^3 - 30*n^2 + 16*n. (End)
E.g.f.: exp(x)*x*(1 + 15*x + 15*x^2). - Stefano Spezia, Jul 15 2024

a(21)-a(33) from Sean A. Irvine, Jul 14 2024

a(34)-a(37) from Alois P. Heinz, Jul 14 2024

A333885 Number of triples (i,j,k) with 1 <= i < j < k <= n such that i divides j divides k.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 6, 7, 9, 9, 16, 16, 18, 20, 26, 26, 33, 33, 40, 42, 44, 44, 59, 60, 62, 65, 72, 72, 84, 84, 94, 96, 98, 100, 119, 119, 121, 123, 138, 138, 150, 150, 157, 164, 166, 166, 192, 193, 200, 202, 209, 209, 224, 226, 241, 243, 245, 245, 276, 276

Offset: 1

Derek Lim, Apr 08 2020

nonn

			The a(4) = 1 triple is (1,2,4).
The a(8) = 6 triples are (1,2,4), (1,2,6), (1,2,8), (1,3,6), (1,4,8), (2,4,8).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000005, A002541, A320224.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
       add(tau(d)-1, d=divisors(n) minus {n}))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 09 2020

a[n_] := a[n] = If[n == 0, 0, a[n - 1] + Sum[DivisorSigma[0, d] - 1, {d, Most @ Divisors[n]}]];
Array[a, 80] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

an = len([(i,j,k) for i in range(1,n+1) for j in range(i+1,n+1) for k in range(j+1,n+1) if j%i==0 and k%j==0])

a(n) = Sum_{m=1..n} Sum_{d|m, dAlois P. Heinz, Apr 09 2020

A327692 Number of length-n phone numbers that can be dialed by a chess knight on a 0-9 keypad that starts on any number and takes n-1 steps.

Original entry on oeis.org

10, 20, 46, 104, 240, 544, 1256, 2848, 6576, 14912, 34432, 78080, 180288, 408832, 944000, 2140672, 4942848, 11208704, 25881088, 58689536, 135515136, 307302400, 709566464, 1609056256, 3715338240, 8425127936, 19453763584

Offset: 1

Derek Lim, Sep 22 2019

nonn

The keypad is of the form:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
| * | 0 | # |
+---+---+---+

			For n = 1 the a(1) = 10 numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
For n = 2 the a(2) = 20 numbers are 04, 06, 16, 18, 27, 29, 34, 38, 43, 49, 40, 61, 67, 60, 72, 76, 81, 83, 92, 94.

Derek Lim, Table of n, a(n) for n = 1..2500

Cf. A280594, A169696.

def number_dialable(N):
    reach = ((4,6),(6,8),(7,9),(4,8),(3,9,0),(),(1,7,0),(2,6),(1,3),(2,4))
    M = [[0] * 10 for _ in range(N)]
    M[0] = [1]*10
    for step in range(1,N):
        for tile in range(10):
            for nxt in reach[tile]:
                M[step][nxt] += M[step-1][tile]
    return [sum(row) for row in M]

Conjectures from Colin Barker, Oct 01 2019: (Start)
G.f.: 2*x*(5 + 10*x - 7*x^2 - 8*x^3 + 2*x^4) / (1 - 6*x^2 + 4*x^4).
a(n) = 6*a(n-2) - 4*a(n-4) for n>6. (End)
Comments from Francesca Arici, Apr 17 2024: (Start)
The recursive formula a(n) = 6*a(n-2) - 4*a(n-4) also holds for n=6.
It can be proved using results from graph theory. Indeed, if we consider the directed graph associated to the knight dialler problem, then a(n) equals the number of paths in the graph of length n-1 in the graph. This number can be expressed in terms of the grand sum of powers of the incidence matrix A(i,j) of the graph.
Moreover, the matrix A is diagonalizable over the reals, with one zero eigenvalue, say L(0)=0. Combining this with the formula for the grand sum of a diagonalizable matrix in term of its eigenvalues, the above conjecture reduces to checking an algebraic condition on the nonzero eigenvalues L(1), ..., L(8) of A. (End)

A327151 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation, such that both permutations in a representative pair are of the same conjugacy class in A_n.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 82, 452, 2369, 18356, 143308, 1396222, 13000455, 152886068

Offset: 0

Derek Lim, Aug 23 2019

			For n = 3, representatives of the a(3) = 3 orbits are: (e,e), ((123),(123)), ((132),(132)), where e is the identity.

Cf. A110143, A327014, A327150.

A327015 Number of equivalence classes of pairs of permutations in S_n where two pairs are equivalent if they are simultaneously conjugate to each other or simultaneously conjugate to each other after a reversal of one pair.

Original entry on oeis.org

1, 1, 3, 8, 28, 98, 518, 3096, 23415, 201795, 1973189, 21347935, 253282652, 3263902430

Offset: 0

Derek Lim, Aug 13 2019

			For n = 2, representatives of the a(2) = 3 classes are: (e,e), (e, (12)), ((12),(12)), where e is identity.

Amit Harlev, Charles R. Johnson, and Derek Lim, The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach, arXiv:1908.03647 [math.SP], 2019.

Cf. A110143, A327014.

a(n) = (A110143(n) + A327014(n)) / 2.

A327150 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation.

Original entry on oeis.org

1, 1, 1, 9, 22, 77, 400, 2624, 20747, 183544, 1826374, 20045348, 240262047, 3120641718, 43665293393, 654731266933, 10472819759734, 178001257647196, 3203520381407270, 60859480965537820, 1217072840308660049

Offset: 0

Derek Lim, Aug 23 2019

nonn

			For n = 3, representatives of the n=9 orbits are (e,e), (e,(123)), (e,(132)), ((123),e), ((132),e), ((123),(123)), ((123),(132)), ((132),(123)), ((132),(132)), where e is the identity.

Derek Lim, Table of n, a(n) for n = 0..61
MathOverflow, A general formula for the number of conjugacy classes of S_n×S_n acted on by S_n

Cf. A000702, A110143, A327014, A327151.

G:= AlternatingGroup(n);; Size(G)*Sum(List(ConjugacyClasses(G), K -> 1/Size(K)));

a(n) = (n!/2) * Sum_{K conjugacy class in A_n} 1/|K|.

A327014 Number of orbits of Sym(n)^2 where Sym(n) acts by conjugation such that both permutations in a representative pair have the same cycle type.

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 135, 613, 3624, 25230, 203640, 1842350, 18535683, 204650313

Offset: 0

Derek Lim, Aug 13 2019

			For n = 2, representatives of the a(2) = 2 orbits are: (e,e), ((12), (12)), where e is identity.

Amit Harlev, Charles R. Johnson, and Derek Lim, The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach, arXiv:1908.03647 [math.SP], 2019.

Cf. A110143, A327015.

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User: Derek Lim

A370649 Dimension of space of equivariant linear maps from R^{n^3} to R^{n^3} under diagonal action of {-1, 1}^n.

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A333885 Number of triples (i,j,k) with 1 <= i < j < k <= n such that i divides j divides k.

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A327692 Number of length-n phone numbers that can be dialed by a chess knight on a 0-9 keypad that starts on any number and takes n-1 steps.

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A327151 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation, such that both permutations in a representative pair are of the same conjugacy class in A_n.

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A327015 Number of equivalence classes of pairs of permutations in S_n where two pairs are equivalent if they are simultaneously conjugate to each other or simultaneously conjugate to each other after a reversal of one pair.

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A327150 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation.

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GAP

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A327014 Number of orbits of Sym(n)^2 where Sym(n) acts by conjugation such that both permutations in a representative pair have the same cycle type.

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