cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A092271 Triangle read by rows. First in a series of triangular arrays counting permutations of partitions.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 8, 6, 1, 24, 30, 20, 10, 1, 120, 144, 90, 40, 15, 1, 720, 840, 504, 210, 70, 21, 1, 5040, 5760, 3360, 1344, 420, 112, 28, 1, 40320, 45360, 25920, 10080, 3024, 756, 168, 36, 1, 362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45, 1, 3628800, 3991680, 2217600, 831600, 237600, 55440, 11088, 1980, 330, 55, 1
Offset: 1

Views

Author

Alford Arnold, Feb 14 2004

Keywords

Comments

Generate signatures in accordance with A086141. Map to partitions in accordance with A025487. Calculate the number of permutations in accordance with Abramowitz and Stegun, p. 831 (reference M2). Display the results as illustrated by A090774. The second array is:
3
20 15
90 120 45
504 630 420 105
...
Apart from the main diagonal, appears to be the same as A211603 (see also A238363) and is related to the infinitesimal generator of A008290; if so, the off-diagonal triangle entries are given by binomial(n,k)*(n-k-1)! for n >= 2 and 0 <= k <= n-2. - Peter Bala, Feb 13 2017
Let aut(p) denote the size of the centralizer of the partition p (see A339016 for the definition). Then row(n) = [n!/aut(p) for p in P], where P are the partitions of n with largest part k and length n + 1 - k. Row sums are A121726. - Peter Luschny, Nov 19 2020

Examples

			The triangle begins:
1:    1
2:    1   1
3:    2   3  1
4:    6   8  6  1
5:   24  30 20 10  1
6:  120 144 90 40 15 1
  ...
From _Peter Luschny_, Nov 19 2020: (Start):
The combinatorial interpretation is illustrated by this computation of row 6:
6! / aut([6])                = 720 / A339033(6, 1) = 720/6   = 120 = T(6, 1)
6! / aut([5, 1])             = 720 / A339033(6, 2) = 720/5   = 144 = T(6, 2)
6! / aut([4, 1, 1])          = 720 / A339033(6, 3) = 720/8   =  90 = T(6, 3)
6! / aut([3, 1, 1, 1])       = 720 / A339033(6, 4) = 720/18  =  40 = T(6, 4)
6! / aut([2, 1, 1, 1, 1])    = 720 / A339033(6, 5) = 720/48  =  15 = T(6, 5)
6! / aut([1, 1, 1, 1, 1, 1]) = 720 / A339033(6, 6) = 720/720 =   1 = T(6, 6)
-------------------------------------------------------------------------------
                                                         Sum:  410 = A121726(6)
(End)
		

References

  • Abramowitz and Stegun, p. 831.

Crossrefs

Programs

  • Mathematica
    f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!];Table[Append[Map[f, Select[Partitions[n], Count[#, Except[1]] == 1 &]], 1], {n,1, 10}] // Grid (* Geoffrey Critzer, Nov 07 2015 *)
  • SageMath
    def A092271(n, k):
        if n == k: return 1
        return factorial(n) // ((n + 1 - k)*factorial(k - 1))
    for n in (1..9): print(n, [A092271(n, k) for k in (1..n)])
    def A092271Row(n):
        if n == 0: return [1]
        f = factorial(n); S = []
        for k in range(n,0,-1):
            for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
                S.append(f // p.aut())
        return S
    for n in (1..9): print(A092271Row(n)) # Peter Luschny, Nov 20 2020

Extensions

More terms from Geoffrey Critzer, Nov 10 2015

A091307 a(n)=6*2^n+4 (Bode Number A003461(n+2)) except for a(1)=6.

Original entry on oeis.org

6, 28, 52, 100, 196, 388, 772, 1540, 3076, 6148, 12292, 24580, 49156, 98308, 196612, 393220, 786436, 1572868, 3145732, 6291460, 12582916, 25165828, 50331652, 100663300, 201326596, 402653188, 805306372, 1610612740, 3221225476, 6442450948, 12884901892
Offset: 1

Views

Author

Alford Arnold, Feb 21 2004

Keywords

Comments

Sequence similar to Bode Numbers relevant to A079946 and numeric partitions.
A053445 describes certain partitions which start triangular arrays of all other numeric partitions; e.g. - 22, 33, 222, 44, 332, 2222, ... A079946 provides the indices for these partitions. (cf. A090324 and A090774).
By expanding the terms of a(n) in a similar manner, the vertex partitions can be readily indexed by noting that the indices increase by eight as follows: 6 28 (one case), 52 60 (two cases), 100 108 116 124 (four cases), 196 204 212 220 228 236 244 252 (eight cases), 388 ...

Examples

			a(3) = 52 because we can write 52 = 2*28 - 4.
		

Crossrefs

Except for initial term, same as A003461(n+2). Cf. A053445, A079946, A090774.

Programs

  • Mathematica
    CoefficientList[Series[2x (3+5x-10x^2)/((1-x)(1-2x)),{x,0,30}],x] (* or *) LinearRecurrence[{3,-2},{0,6,28,52},40] (* Harvey P. Dale, Sep 01 2021 *)

Formula

a(1) = 6, a(2) = 28, a(n) = 2*a(n-1) - 4 for n > 2.
G.f.: 2*x*(3+5*x-10*x^2)/((1-x)*(1-2*x)). - Colin Barker, Mar 12 2012

Extensions

Edited by M. F. Hasler, Apr 07 2009
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