A244490 -id:A244490 - cp's OEIS frontend

A244491 Number of minimal idempotent generating sets for the singular part P_n \ S_n of the partition monoid P_n.

1, 1, 3, 20, 201, 2604, 40915, 754368, 15960945, 381141008, 10139372451, 297356237760, 9530800099513, 331453265976000, 12430323314648499, 500046099516905984, 21478615942550889825, 981110493372418629888, 47489191763845877910595

Offset: 0

N. J. A. Sloane, Jul 05 2014

nonn

J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014-2016.

A038205 := proc(n)
    option remember ;
    if n = 0 then
        1;
    elif n <=2 then
        0 ;
    else
        (n-1)*procname(n-1)+(n-1)*(n-2)*procname(n-3) ;
    end if;
end proc:
A244490 := proc(n,k)
    add((-1)^i*binomial(k,2*i)*doublefactorial(2*i-1)*n^(k-2*i),i=0..floor(k/2)) ;
end proc:
A244491 := proc(n)
    add(binomial(n,k)*A038205(k)*A244490(n,n-k),k=0..n) ;
end proc:
seq(A244491(n),n=0..30) ; # R. J. Mathar, Aug 26 2014

a05[n_] := SeriesCoefficient[Exp[-x - x^2/2]/(1 - x), {x, 0, n}]*n!;
a90[n_, k_] := Sum[(-1)^i*Binomial[k, 2i]*(2i-1)!!*n^(k-2*i), {i, 0, k/2}];
a[n_] := Sum[Binomial[n, k]*a05[k]*a90[n, n - k], {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 01 2017, after R. J. Mathar *)

An explicit formula is given in Th. 7.13 of East-Gray.

A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k] exp(nx)*sech(x), n>=0, k>=0.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, 0, 0, 2, 1, 5, -2, 3, 3, 1, 0, 0, 2, 8, 4, 1, -61, 16, -3, 18, 15, 5, 1, 0, 0, 2, 32, 52, 24, 6, 1, 1385, -272, 63, 48, 165, 110, 35, 7, 1, 0, 0, 2, 128, 484, 480, 198, 48, 8, 1, -50521, 7936, -1383, 528, 1395, 2000, 1085, 322, 63, 9, 1

Offset: 0

Peter Luschny, Dec 14 2014

This two-dimensional array of numbers can be seen as a generalization of the Euler secant and Euler tangent numbers (which are in their compressed and signless form A000364 resp. A000182 or interleaved in A000111). The cases n=0 and n=1 reduce to their expanded and signed forms A122045 and A155585. Moreover the columns are the values of the Swiss-Knife polynomials A153641 evaluated at the nonnegative integers.

Subsequences [3,3,1], [8,4,1], [15,5,1], [24,6,1], [35,7,1], [48,8,1], [63,9,1] found in rows of this entry, as a triangular array, are present in the antidiagonals of Table 5 of the East and Gray reference (A244490), and some subsequences in the rows of Table 5 are found in the antidiagonals of this entry, including [3,2,1] and [1,1]. Equivalently, the first four columns of Table 5 are embedded in this entry viewed as a square array on the table page. An explicit formula with combinatorial interpretations for these numbers is provided in the reference, and others are known for the corresponding columns for the modified Hermite polynomials of A244490. - Tom Copeland, Oct 04 2016

			Square array starts:
  [n\k][0][1] [2]  [3]   [4]   [5]    [6]     [7]     [8]
  [0]   1, 0, -1,   0,    5,    0,   -61,      0,   1385, ... A122045
  [1]   1, 1,  0,  -2,    0,   16,     0,   -272,      0, ... A155585
  [2]   1, 2,  3,   2,   -3,    2,    63,      2,  -1383, ... A119880
  [3]   1, 3,  8,  18,   32,   48,   128,    528,    512, ... A119881
  [4]   1, 4, 15,  52,  165,  484,  1395,   4372,  14505, ...
  [5]   1, 5, 24, 110,  480, 2000,  8064,  32240, 130560, ... A225116
  [6]   1, 6, 35, 198, 1085, 5766, 29855, 151878, 766745, ...
  A000012, A001477, A067998, A121670, ...
Triangular array starts:
                1,
              0,  1,
           -1,  1,  1,
          0,  0,  2,  1,
        5, -2,  3,  3,  1,
      0,  0,  2,  8,  4,  1,
  -61, 16, -3, 18, 15,  5,  1.

J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

Cf. A122045, A155585, A119880, A119881, A225116, A000364, A000182, A000111, A067998, A121670, A153641, A247501.

Cf. A244490.

# EGF (row)
egf := n -> exp(n*x)*sech(x):
seq(print(seq(k!*coeff(series(egf(n),x,k+2),x,k),k=0..8)), n=0..6);
# Swiss-Knife polynomial (column)
SKP := proc(n, x) local v, k, A; A := k -> `if`(k mod 4 = 0,0,(-1)^iquo(k,4)); add(2^iquo(-k,2)*A(k+1)*add((-1)^v* binomial(k,v)*(v+x+1)^n,v=0..k), k=0..n); expand(%) end:
seq(print(seq(SKP(k, n), n=0..9)), k=0..6);
# OGF (column)
col := proc(n, len) local T; T := A247501_row(n);
(-1)^(n+1)*add(T[k+1]/(x-1)^(k+1),k=0..n);
seq(coeff(series(%,x,len+1),x,j),j=0..len) end:
seq(print(col(n,8)), n=0..6);

nmax = 10; Clear[row]; row[n_] := row[n] = CoefficientList[Exp[n*x]*Sech[x] + O[x]^(nmax+2), x][[1 ;; nmax+1]]*Range[0, nmax]!;
rows = Table[row[n], {n, 0, nmax}];
T[n_, k_] := rows[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 03 2017 *)

G.f. for column k: the k-th column consists of the values of the k-th Swiss-Knife polynomial skp_{k}(x) evaluated at x = 0,1,2,...

O.g.f. for column k: Sum_{j=0..k} (-1)^(k+1)*A247501(k,j)/(x-1)^(j+1).

A276999 Triangle read by rows, T(n,k) = n^k - 2^(k/2)*KummerU(-k/2,1/2,n^2/2) for 0<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 9, 0, 0, 1, 12, 93, 0, 0, 1, 15, 147, 1175, 0, 0, 1, 18, 213, 2070, 17835, 0, 0, 1, 21, 291, 3325, 33825, 317667, 0, 0, 1, 24, 381, 5000, 58575, 635208, 6506647, 0, 0, 1, 27, 483, 7155, 94785, 1164429, 13536453, 150776397

Offset: 0

Peter Luschny, Oct 06 2016

East and Gray (p. 24) give a combinatorial interpretation of the numbers: A function f: Y -> X with Y <= X (<= inclusion) has a 2-cycle if there exists x, y in Y with x != y, f(x) = y and f(y) = x. Then T(n,k) = card({f: [k] -> [n]: f has 2-cycles}).

			Triangle begins:
0;
0, 0;
0, 0, 1;
0, 0, 1,  9;
0, 0, 1, 12,  93;
0, 0, 1, 15, 147, 1175;
0, 0, 1, 18, 213, 2070, 17835;
0, 0, 1, 21, 291, 3325, 33825,  317667;
0, 0, 1, 24, 381, 5000, 58575,  635208,  6506647;
0, 0, 1, 27, 483, 7155, 94785, 1164429, 13536453, 150776397;
.
For instance T(3,3) = 9 because there are 27 functions [3]->[3], 18 of which have
no 2-cycles. The 9 functions which have 2-cycles are (represented as [f(1), f(2),
f(3)]): [1, 3, 2], [2, 1, 1], [2, 1, 2], [2, 1, 3], [2, 3, 2], [3, 1, 1],
[3, 2, 1], [3, 3, 1], [3, 3, 2].

J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

T(n,k) = n^k - A244490(n,k), T(n,3) = A008585(n) for n>=3, T(n,4) = A224334(n-1) for n>=4, T(n,5) = A127694(n-3) for n>=5.

T := (n,k) -> n^k - 2^(k/2)*KummerU(-k/2, 1/2, n^2/2):
seq(seq(simplify(T(n,k)), k=0..n), n=0..9);

Table[Simplify[n^k - 2^(-k/2) HermiteH[k, n/Sqrt[2]]], {n, 0, 10}, {k, 0, n}] // Flatten

def T(n, k):
    @cached_function
    def h(n, x):
        if n == 0: return 1
        if n == 1: return 2*x
        return 2*(x*h(n-1,x)-(n-1)*h(n-2,x))
    return n^k - h(k, n/sqrt(2))/2^(k/2)
for n in range(10):
    print([T(n,k) for k in (0..n)])

T(n,k) = n^k - 2^(-k/2)*HermiteH(k, n/sqrt(2)).
T(n,k) = n^k - Sum_{i=0..k/2} k!/((-2)^i*i!*(k-2*i)!)*n^(k-2*i).
T(n,k) = n^k*(1-hypergeom([-k/2, (1-k)/2], [], -2/n^2)) for n>=1.
T(n,k) ~ n^k*(((k-1)*k)/(2*n^2)-(k*(k^3-6*k^2+11*k-6))/(8*n^4)+(k*(k^5-15*k^4 +85*k^3-225*k^2+274*k-120))/(48*n^6)+O((1/n)^7)).

cp's OEIS Frontend

A244491 Number of minimal idempotent generating sets for the singular part P_n \ S_n of the partition monoid P_n.

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A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k] exp(nx)*sech(x), n>=0, k>=0.

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A276999 Triangle read by rows, T(n,k) = n^k - 2^(k/2)*KummerU(-k/2,1/2,n^2/2) for 0<=k<=n.

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cp's OEIS Frontend

A244491 Number of minimal idempotent generating sets for the singular part P_n \ S_n of the partition monoid P_n.

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Maple

Mathematica

Formula

A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k!*[x^k] exp(n*x)*sech(x), n>=0, k>=0.

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Maple

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A276999 Triangle read by rows, T(n,k) = n^k - 2^(k/2)*KummerU(-k/2,1/2,n^2/2) for 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k] exp(nx)*sech(x), n>=0, k>=0.