A092271 -id:A092271 - cp's OEIS frontend

A008290 Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 9, 8, 6, 0, 1, 44, 45, 20, 10, 0, 1, 265, 264, 135, 40, 15, 0, 1, 1854, 1855, 924, 315, 70, 21, 0, 1, 14833, 14832, 7420, 2464, 630, 112, 28, 0, 1, 133496, 133497, 66744, 22260, 5544, 1134, 168, 36, 0, 1, 1334961, 1334960, 667485, 222480, 55650, 11088, 1890, 240, 45, 0, 1

Offset: 0

N. J. A. Sloane

This is a binomial convolution triangle (Sheffer triangle) of the Appell type: (exp(-x)/(1-x),x), i.e., the e.g.f. of column k is (exp(-x)/(1-x))*(x^k/k!). See the e.g.f. given by V. Jovovic below. - Wolfdieter Lang, Jan 21 2008
The formula T(n,k) = binomial(n,k)*A000166(n-k), with the derangements numbers (subfactorials) A000166 (see also the Charalambides reference) shows the Appell type of this triangle. - Wolfdieter Lang, Jan 21 2008
T(n,k) is the number of permutations of {1,2,...,n} having k pairs of consecutive right-to-left minima (0 is considered a right-to-left minimum for each permutation). Example: T(4,2)=6 because we have 1243, 1423, 4123, 1324, 3124 and 2134; for example, 1324 has right-to-left minima in positions 0-1,3-4 and 2134 has right-to-left minima in positions 0,2-3-4, the consecutive ones being joined by "-". - Emeric Deutsch, Mar 29 2008
T is an example of the group of matrices outlined in the table in A132382--the associated matrix for the sequence aC(0,1). - Tom Copeland, Sep 10 2008
A refinement of this triangle is given by A036039. - Tom Copeland, Nov 06 2012
This triangle equals (A211229(2*n,2*k)) n,k >= 0. - Peter Bala, Dec 17 2014

			exp((y-1)*x)/(1-x) = 1 + y*x + (1/2!)*(1+y^2)*x^2 + (1/3!)*(2 + 3*y + y^3)*x^3 + (1/4!)*(9 + 8*y + 6*y^2 + y^4)*x^4 + (1/5!)*(44 + 45*y + 20*y^2 + 10*y^3 + y^5)*x^5 + ...
Triangle begins:
       1
       0      1
       1      0     1
       2      3     0     1
       9      8     6     0    1
      44     45    20    10    0    1
     265    264   135    40   15    0   1
    1854   1855   924   315   70   21   0  1
   14833  14832  7420  2464  630  112  28  0 1
  133496 133497 66744 22260 5544 1134 168 36 0 1
...
From _Peter Bala_, Feb 13 2017: (Start)
The infinitesimal generator has integer entries given by binomial(n,k)*(n-k-1)! for n >= 2 and 0 <= k <= n-2 and begins
   0
   0  0
   1  0  0
   2  3  0  0
   6  8  6  0 0
  24 30 20 10 0 0
...
It is essentially A238363 (unsigned and omitting the main diagonal), A211603 (with different offset) and appears to be A092271, again without the main diagonal. (End)

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 173, Table 5.2 (without row n=0 and column k=0).
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194.
Arnold Kaufmann, Introduction à la combinatorique en vue des applications, Dunod, Paris, 1968. See p. 92.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from T. D. Noe)
Taha Akbari, Prove using combinatorics Sum_{k=0..n} (k-1)^2 D_n(k)=n!, Mathematics Stack Exchange, Jun 06 2017
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Stefano Capparelli, Margherita Maria Ferrari, Emanuele Munarini, and Norma Zagaglia Salvi, A Generalization of the "Problème des Rencontres", J. Int. Seq. 21 (2018), #18.2.8.
Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016. See Table 1.
S. K. Das and N. Deo, Rencontres graphs: a family of bipartite graphs, Fib. Quart., Vol. 25, No. 3, August 1987, 250-262.
Robert W. Donley Jr, Binomial arrays and generalized Vandermonde identities, arXiv:1905.01525 [math.CO], 2019.
FindStat - Combinatorial Statistic Finder, The number of adjacencies of a permutation, 0 appended, The number of fixed points of a permutation
I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.
J. Liese and J. Remmel, Q-analogues of the number of permutations with k-excedances, PU. M. A. Vol. 21 (2010), No. 2, pp. 285-320 (see E_{n,0}(x) in Table 1 p. 291).
L. Takacs, On the "problème des ménages", Discr. Math. 36 (3) (1981) 289-297, Table 2.
Wikipedia, Rencontres numbers.
Index entries for sequences related to permutations with fixed points

Mirror of triangle A098825.
Columns give A000166, A000240, A000387, A000449, A000475, A129135, A129136, A129149, A129153, A129217, A129218, A129238, A129255.
Cf. A055137, A008291, A098558.
Cf. A080955.
Cf. A000012, A000142 (row sums), A000354.
Cf. A170942. Sub-triangle of A211229.
Cf. A092271, A111492, A211603, A238363.
T(2n,n) gives A281262.
Cf. A000023, A133314, A238385, A320582, A335111.

a008290 n k = a008290_tabl !! n !! k
a008290_row n = a008290_tabl !! n
a008290_tabl = map reverse a098825_tabl
-- Reinhard Zumkeller, Dec 16 2013

T:= proc(n,k) T(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
      (T(n-1, 0)+T(n-2, 0))), binomial(n, k)*T(n-k, 0))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Mar 15 2013

a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 size = 8; Table[Binomial[n, k]a[n - k], {n, 0, size}, {k, 0, n}] // TableForm (* Harlan J. Brothers, Mar 19 2007 *)
T[n_, k_] := Subfactorial[n-k]*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2017 *)
T[n_, k_] := If[n<1, Boole[n==0 && k==0], T[n, k] = T[n-1, k-1] + T[n-1, k]*(n-1-k) + T[n-1, k+1]*(k+1)]; (* Michael Somos, Sep 13 2024 *)
T[0, 0]:=1; T[n_, 0]:=T[n, 0]=n  T[n-1, 0]+(-1)^n; T[n_, k_]:=T[n, k]=n/k T[n-1, k-1];
Flatten@Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Oliver Seipel, Nov 26 2024 *)

{T(n, k) = if(k<0 || k>n, 0, n!/k! * sum(i=0, n-k, (-1)^i/i!))}; /* Michael Somos, Apr 26 2000 */

T(n, k) = T(n-1, k)*n + binomial(n, k)*(-1)^(n-k) = T(n, k-1)/k + binomial(n, k)*(-1)^(n-k)/(n-k+1) = T(n-1, k-1)*n/k = T(n-k, 0)*binomial(n, k) = A000166(n-k)*binomial(n,k) [with T(0, 0) = 1]; so T(n, n) = 1, T(n, n-1) = 0, T(n, n-2) = n*(n-1)/2 for n >= 0.
Sum_{k=0..n} T(n, k) = Sum_{k=0..n} k * T(n, k) = n! for all n > 0, n, k integers. - Wouter Meeussen, May 29 2001
From Vladeta Jovovic, Aug 12 2002: (Start)
O.g.f. for k-th column: (1/k!)*Sum_{i>=k} i!*x^i/(1+x)^(i+1).
O.g.f. for k-th row: k!*Sum_{i=0..k} (-1)^i/i!*(1-x)^i. (End)
E.g.f.: exp((y-1)*x)/(1-x). - Vladeta Jovovic, Aug 18 2002
E.g.f. for number of permutations with exactly k fixed points is x^k/(k!*exp(x)*(1-x)). - Vladeta Jovovic, Aug 25 2002
Sum_{k=0..n} T(n, k)*x^k is the permanent of the n X n matrix with x's on the diagonal and 1's elsewhere; for x = 0, 1, 2, 3, 4, 5, 6 see A000166, A000142, A000522, A010842, A053486, A053487, A080954. - Philippe Deléham, Dec 12 2003; for x = 1+i see A009551 and A009102. - John M. Campbell, Oct 11 2011
T(n, k) = Sum_{j=0..n} A008290(n, j)*k^(n-j) is the permanent of the n X n matrix with 1's on the diagonal and k's elsewhere; for k = 0, 1, 2 see A000012, A000142, A000354. - Philippe Deléham, Dec 13 2003
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j,k)*n!/j!. - Paul Barry, May 25 2006
T(n,k) = (n!/k!)*Sum_{j=0..n-k} ((-1)^j)/j!, 0 <= k <= n. From the Appell type of the triangle and the subfactorial formula.
T(n,0) = n*Sum_{j=0..n-1} (j/(j+1))*T(n-1,j), T(0,0)=1. From the z-sequence of this Sheffer triangle z(j)=j/(j+1) with e.g.f. (1-exp(x)*(1-x))/x. See the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jan 21 2008
T(n,k) = (n/k)*T(n-1,k-1) for k >= 1. See above. From the a-sequence of this Sheffer triangle a(0)=1, a(n)=0, n >= 1 with e.g.f. 1. See the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jan 21 2008
From Henk P. J. van Wijk, Oct 29 2012: (Start)
T(n,k) =              T(n-1,k)*(n-1-k) + T(n-1,k+1)*(k+1) for k=0 and
T(n,k) = T(n-1,k-1) + T(n-1,k)*(n-1-k) + T(n-1,k+1)*(k+1) for k>=1.
(End)
T(n,k) = A098825(n,n-k). - Reinhard Zumkeller, Dec 16 2013
Sum_{k=0..n} k^2 * T(n, k) = 2*n! if n > 1. - Michael Somos, Jun 06 2017
From Tom Copeland, Jul 26 2017: (Start)
The lowering and raising operators of this Appell sequence of polynomials P(n,x) are L = d/dx and R = x + d/dL log[exp(-L)/(1-L)] = x-1 + 1/(1-L) = x + L + L^2 - ... such that L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x).
P(n,x) = (1-L)^(-1) exp(-L) x^n = (1+L+L^2+...)(x-1)^n = n! Sum_{k=0..n} (x-1)^k / k!.
The formalism of A133314 applies to the pair of entries A008290 and A055137.
The polynomials of this pair P_n(x) and Q_n(x) are umbral compositional inverses; i.e., P_n(Q.(x)) = x^n = Q_n(P.(x)), where, e.g., (Q.(x))^n = Q_n(x).
For more on the infinitesimal generator, noted by Bala below, see A238385. (End)
Sum_{k=0..n} k^m * T(n,k) = A000110(m)*n! if n >= m. - Zhujun Zhang, May 24 2019
Sum_{k=0..n} (k+1) * T(n,k) = A098558(n). - Alois P. Heinz, Mar 11 2022
From Alois P. Heinz, May 20 2023: (Start)
Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).
Sum_{k=0..n} (-1)^k * k * T(n,k) = A335111(n). (End)
T(n,k) = A145224(n,k)+A145225(n,k), refined by even and odd perms. - R. J. Mathar, Jul 06 2023

Comments and more terms from Michael Somos, Apr 26 2000 and Christian G. Bower, Apr 26 2000

A238363 Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.

Original entry on oeis.org

1, -1, 2, 2, -3, 3, -6, 8, -6, 4, 24, -30, 20, -10, 5, -120, 144, -90, 40, -15, 6, 720, -840, 504, -210, 70, -21, 7, -5040, 5760, -3360, 1344, -420, 112, -28, 8, 40320, -45360, 25920, -10080, 3024, -756, 168, -36, 9, -362880, 403200, -226800, 86400, -25200, 6048, -1260, 240, -45, 10

Offset: 1

Tom Copeland, Feb 25 2014

Let D=d/dx and [A,B]=A·B-B·A. Then each row corresponds to the coefficients of the operators :xD:^k = x^k D^k in the expansion of the commutator [log(D),:xD:^n]=[-log(x),:xD:^n]=sum(k=0 to n-1, a(n,k) :xD:^k). The e.g.f. is derived from [log(D), exp(t:xD:)]=[-log(x), exp(t:xD:)]= log(1+t)exp(t:xD:), using the shift property exp(t:xD:)f(x)=f((1+t)x).
The reversed unsigned array is A111492.
See the mathoverflow link and link therein to an associated mathstackexchange question for other formulas for log(D). In addition, R_x = log(D) = -log(x) + c - sum[n=1 to infnty, (-1)^n 1/n :xD:^n/n!]=
-log(x) + Psi(1+xD) = -log(x) + c + Ein(:xD:), where c is the Euler-Mascheroni constant, Psi(x), the digamma function, and Ein(x), a breed of the exponential integrals (cf. Wikipedia). The :xD:^k ops. commute; therefore, the commutator reduces to the -log(x) term.
Also the n-th row corresponds to the expansion of d[(xD)!/(xD-n)!]/d(xD) = d[:xD:^n]/d(xD) in the operators :xD:^k, or, equivalently, the coefficients of x in  d[z!/(z-n)!]/dz=d[St1(n,z)]]/dz evaluated umbrally with z=St2(.,x), i.e., z^n replaced by St2(n,x), where St1(n,x) and St2(n,x) are the signed and unsigned Stirling polynomials of the first (A008275) and second (A008277) kinds. The derivatives of the unsigned St1 are A028421. See examples. This formalism follows from the relations between the raising and lowering operators presented in the MathOverflow link and the Pincherle derivative. The results can be generalized through the operator relations in A094638, which are related to the celebrated Witt Lie algebra and pseudodifferential operators / symbols, to encompass other integral arrays.
A002741(n)*(-1)^(n+1) (row sums), A002104(n)*(-1)^(n+1) (alternating row sums). Column sequences: A133942(n-1), A001048(n-1), A238474, ... - Wolfdieter Lang, Mar 01 2014
Add an additional head row of zeros to the lower triangular array and denote it as T (with initial indexing in columns and rows being 0). Let dP = A132440, the infinitesimal generator for the Pascal matrix, and I, the identity matrix, then exp(T)=I+dP, i.e., T=log(I+dP). Also, (T_n)^n=0, where T_n denotes the n X n submatrix, i.e., T_n is nilpotent of order n. - Tom Copeland, Mar 01 2014
Any pair of lowering and raising ops. L p(n,x) = n·p(n-1,x) and R p(n,x) = p(n+1,x) satisfy [L,R]=1 which implies (RL)^n = St2(n,:RL:), and since (St2(·,u))!/(St2(·,u)-n)!= u^n, when evaluated umbrally, d[(RL)!/(RL-n)!]/d(RL) = d[:RL:^n]/d(RL) is well-defined and gives A238363 when the LHS is reduced to a sum of :RL:^k terms, exactly as for L=d/dx and R=x above. (Note that R_x above is a raising op. different from x, with associated L_x=-xD.) - Tom Copeland, Mar 02 2014
For relations to colored forests, disposition of flags on flagpoles, and the colorings of the vertices of the complete graphs K_n, encoded in their chromatic polynomials, see A130534. - Tom Copeland, Apr 05 2014
The unsigned triangle, omitting the main diagonal, gives A211603. See also A092271. Related to the infinitesimal generator of A008290. - Peter Bala, Feb 13 2017

			The first few row polynomials are
p(1,x)=  1
p(2,x)= -1 + 2x
p(3,x)=  2 - 3x + 3x^2
p(4,x)= -6 + 8x - 6x^2 + 4x^3
p(5,x)= 24 -30x +20x^2 -10x^3 + 5x^4
...........
For n=3: z!/(z-3)!=z^3-3z^2+2z=St1(3,z) with derivative 3z^2-6z+2, and
3·St2(2,x)-6·St2(1,x)+2=3(x^2+x)-6x+2=3x^2-3x+2=p(3,x). To see the relation to the operator formalism, note that (xD)^k=St2(k,:xD:) and (xD)!/(xD-k)!=[St2(·,:xD:)]!/[St2(·,:xD:)-k]!= :xD:^k.
The triangle a(n,k) begins:
n\k       0       1       2      3      4     5      6    7   8   9 ...
1:        1
2:       -1       2
3:        2      -3       3
4:       -6       8      -6      4
5:       24     -30      20    -10      5
6:     -120     144     -90     40    -15     6
7:      720    -840     504   -210     70   -21      7
8:    -5040    5760   -3360   1344   -420   112    -28    8
9:    40320  -45360   25920 -10080   3024  -756    168  -36   9
10: -362880  403200 -226800  86400 -25200  6048  -1260  240 -45  10
... formatted by _Wolfdieter Lang_, Mar 01 2014
-----------------------------------------------------------------------

Tom Copeland, Riemann zeta function at positive integers and an Appell sequence of polynomials, 2012.
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator, 2014.
Tom Copeland, Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering, 2014.
Tom Copeland, Compositional inverse operators and Sheffer sequences, 2016.

Cf. A094587, A132013, A132014, A132159, A235706, A238385.

Cf. A008290, A092271, A211603.

a[n_, k_] := (-1)^(n-k-1)*n!/((n-k)*k!); Table[a[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 09 2015 *)

a(n,k) = (-1)^(n-k-1)*n!/((n-k)*k!) for k=0 to (n-1).
E.g.f.: log(1+t)*exp(x*t).
E.g.f.for unsigned array: -log(1-t)*exp(x*t).
The lowering op. for the row polynomials is L=d/dx, i.e., L p(n,x) = n*p(n-1,x).
An e.g.f. for an unsigned related version is -log(1+t)*exp(x*t)/t= exp(t*s(·,x)) with s(n,x)=(-1)^n * p(n+1,-x)/(n+1). Let L=d/dx and R= x-(1/((1-D)log(1-D))+1/D),then R s(n,x)= s(n+1,x) and L s(n,x)= n*s(n-1,x), defining a special Sheffer sequence of polynomials, an Appell sequence. So, R (-1)^(n-1) p(n,-x)/n = (-1)^n p(n+1,-x)/(n+1).
From Tom Copeland, Apr 17 2014: (Start)
Dividing each diagonal by its first element (-1)^(n-1)*(n-1)! yields the reverse of A104712.
Multiply each n-th diagonal of the Pascal lower triangular matrix by x^n and designate the result as A007318(x) = P(x). Then with dP = A132440, M = padded A238363 = A238385-I, I = identity matrix, and (B(.,x))^n = B(n,x) = the n-th Bell polynomial Bell(n,x) of A008277,
A) P(x)= exp(x*dP) = exp[x*(e^M-I)] = exp[M*B(.,x)] = (I+dP)^B(.,x), and
B) P(:xD:)=exp(dP:xD:)=exp[(e^M-I):xD:]=exp[M*B(.,:xD:)]=exp[M*xD]=
  (1+dP)^(xD) with action P(:xD:)g(x) = exp(dP:xD:)g(x) = g[(I+dP)*x].
C) P(x)^m = P(m*x). P(2x) = A038207(x) = exp[M*B(.,2x)], face vectors of n-D hypercubes. (End)
From Tom Copeland, Apr 26 2014: (Start)
M = padded A238363 = A238385-I
A)  = [St1]*[dP]*[St2] = [padded A008275]*A132440*A048993
B)  = [St1]*[dP]*[St1]^(-1)
C)  = [St2]^(-1)*[dP]*[St2]
D)  = [St2]^(-1)*[dP]*[St1]^(-1),
  where [St1]=padded A008275 just as [St2]=A048993=padded A008277.
E) P(x) = [St2]*exp(x*M)*[St1] = [St2]*(I + dP)^x*[St1].
F) exp(x*M) = [St1]*P(x)*[St2] = (I + dP)^x,
  where (I + dP)^x = sum(k>=0, C(x,k)*dP^k).
Let the row vector Rv=(c0 c1 c2 c3 ...) and the column vector Cv(x)=(1 x x^2 x^3 ...)^Transpose. Form the power series V(x)= Rv * Cv(x) and W(y) := V(x.) evaluated umbrally with (x.)^n = x_n = (y)_n = y!/(y-n)!. Then
G) U(:xD:) = dV(:xD:)/d(xD) = dW(xD)/d(xD) evaluated with (xD)^n = Bell(n,:xD:),
H) U(x) = dV(x.)/dy := dW(y)/dy evaluated with y^n=y_n=Bell(n,x), and
I) U(x) = Rv * M * Cv(x).   (Cf. A132440, A074909.) (End)
The Bernoulli polynomials Ber_n(x) are related to the polynomials q_n(x) = p(n+1,x) / (n+1) with the e.g.f. [log(1+t)/t] e^(xt) (cf. s_n (x) above) as Ber_n(x) = St2_n[q.(St1.(x))], umbrally, or [St2]*[q]*[St1], in matrix form. Since q_n(x) is an Appell sequence of polynomials, q_n(x) = [log(1+D_x)/D_x]x^n. - Tom Copeland, Nov 06 2016

Pincherle formalism added by Tom Copeland, Feb 27 2014

A339033 Triangle read by rows, T(n, k) for 0 <= k <= n. T(n, 0) = 0^n; T(n, n) = n!; otherwise T(n, k) = (n + 1 - k)*(k - 1)!.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 2, 6, 0, 4, 3, 4, 24, 0, 5, 4, 6, 12, 120, 0, 6, 5, 8, 18, 48, 720, 0, 7, 6, 10, 24, 72, 240, 5040, 0, 8, 7, 12, 30, 96, 360, 1440, 40320, 0, 9, 8, 14, 36, 120, 480, 2160, 10080, 362880, 0, 10, 9, 16, 42, 144, 600, 2880, 15120, 80640, 3628800

Offset: 0

Peter Luschny, Nov 20 2020

Related to the multinomial that is called M2 in Abramowitz and Stegun, p. 831.

			Triangle starts:
0: [1]
1: [0, 1]
2: [0, 2, 2]
3: [0, 3, 2,  6]
4: [0, 4, 3,  4, 24]
5: [0, 5, 4,  6, 12, 120]
6: [0, 6, 5,  8, 18,  48, 720]
7: [0, 7, 6, 10, 24,  72, 240, 5040]
8: [0, 8, 7, 12, 30,  96, 360, 1440, 40320]
9: [0, 9, 8, 14, 36, 120, 480, 2160, 10080, 362880]

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, page 831.

Cf. A339034 (row sums), A092271.

A339033[n_, k_] := Which[k == 0, Boole[n == 0], n == k, n!, True, (n+1-k)*(k-1)!];
Table[A339033[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 31 2024 *)

def  A339033(n, k):
    if k == 0: return 0^n
    if n == k: return factorial(n)
    return (n + 1 - k)*factorial(k - 1)
for n in (0..10): print([A339033(n, k) for k in (0..n)])
def A339033Row(n):
    S = [0^n]
    for k in range(n, 0, -1):
        for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
            S.append(p.aut())
    return S
for n in (0..10): print(A339033Row(n))

T(n, k) = n! / A092271(n, k) for k > 0.

A211603 Triangular array read by rows: T(n,k) is the number of n-permutations that are pure cycles having exactly k fixed points; n>=2, 0<=k<=n-2.

Original entry on oeis.org

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

Offset: 2

Geoffrey Critzer, Feb 10 2013

Equivalently, T(n,k) is the number of n-permutations that are pure cycles of length n-k.
Row sums = A006231.
With a different row and column indexing, this triangle equals the infinitesimal generator of A008290. Equals the unsigned version of A238363, omitting its main diagonal. See also A092271. - Peter Bala, Feb 13 2017

			T(3,1) = 3 because we have (1)(2,3), (2)(1,3), (3)(1,2).
1;
2, 3;
6, 8, 6;
24, 30, 20, 10;
120, 144, 90, 40, 15;
720, 840, 504, 210, 70, 21;
5040, 5760, 3360, 1344, 420, 112, 28;
40320, 45360, 25920, 10080, 3024, 756, 168, 36;
362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45;

Alois P. Heinz, Rows n = 2..150, flattened

Cf. A006231 (row sums), A008290, A092271, A111492, A238363.

T:= (n, k)-> binomial(n, k)*(n-k-1)!:
seq(seq(T(n,k), k=0..n-2), n=2..12);  # Alois P. Heinz, Feb 10 2013

nn=10;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[ Series[Exp[y x](Log[1/(1-x)]-x),{x,0,nn}],{x,y}]]//Grid

E.g.f.: exp(y*x)*(log(1/(1-x))-x).
T(n,k) = C(n,k)*(n-k-1)!. - Alois P. Heinz, Feb 10 2013
T(n,k) = A111492(n,n-k). - R. J. Mathar, Mar 07 2013

A121726 Sum sequence A000522 then subtract 0,1,2,3,4,5,...

Original entry on oeis.org

1, 2, 6, 21, 85, 410, 2366, 16065, 125665, 1112074, 10976174, 119481285, 1421542629, 18348340114, 255323504918, 3809950976993, 60683990530209, 1027542662934898, 18430998766219318, 349096664728623317, 6962409983976703317, 145841989688186383338, 3201192743180799343822

Offset: 1

Alford Arnold, Aug 17 2006

Let aut(p) denote the size of the centralizer of the partition p (see A339016 for the definition). Then a(n) = Sum_{p in P} n!/aut(p), where P are the partitions of n with largest part k and length n + 1 - k. - Peter Luschny, Nov 19 2020

			A000522 begins     1 2 5 16 65 326 ...
with sums          1 3 8 24 89 415 ...
so sequence begins 1 2 6 21 85 410 ...
.
From _Peter Luschny_, Nov 19 2020: (Start):
The combinatorial interpretation is illustrated by this computation of a(5):
5! / aut([5])             = 120 / A339033(5, 1) = 120/5   = 24
5! / aut([4, 1])          = 120 / A339033(5, 2) = 120/4   = 30
5! / aut([3, 1, 1])       = 120 / A339033(5, 3) = 120/6   = 20
5! / aut([2, 1, 1, 1])    = 120 / A339033(5, 4) = 120/12  = 10
5! / aut([1, 1, 1, 1, 1]) = 120 / A339033(5, 5) = 120/120 =  1
--------------------------------------------------------------
                                                Sum: a(5) = 85
(End)

Also the row sums of A092271.

Cf. A000522, A006231, A002104, A339016, A339033.

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n], Count[#, Except[1]] == 1 &]]] + 1, {n, 1, 20}] (* Geoffrey Critzer, Nov 07 2015 *)

A000522(n)={ return( sum(k=0,n,n!/k!)) ; } A121726(n)={ return(sum(k=0,n-1,A000522(k))-n+1) ; } { for(n=1,25, print1(A121726(n),",") ; ) ; } \\ R. J. Mathar, Sep 02 2006

def A121726(n):
    def h(n, k):
        if n == k: return 1
        return factorial(n)//((n + 1 - k)*factorial(k - 1))
    return sum(h(n, k) for k in (1..n))
print([A121726(n) for n in (1..23)])
# Demonstrates the combinatorial view:
def A121726(n):
    if n == 0: return 1
    f = factorial(n); S = 0
    for k in (0..n):
        for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
            S += (f // p.aut())
    return S
print([A121726(n) for n in (1..23)]) # Peter Luschny, Nov 20 2020

a(n) = A006231(n) + 1 = A002104(n) - (n-1). - Franklin T. Adams-Watters, Aug 29 2006

E.g.f.: exp(x)*(log(1/(1-x)) - x + 1). - Geoffrey Critzer, Nov 07 2015

More terms from Franklin T. Adams-Watters, Aug 29 2006

More terms from R. J. Mathar, Sep 02 2006

A339016 A classification of permutations based on their cycle length and the size of the centralizer of their cycle type. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 3, 21, 0, 0, 0, 0, 35, 85, 0, 0, 0, 0, 55, 255, 410, 0, 0, 0, 0, 0, 1015, 1659, 2366, 0, 0, 0, 0, 0, 2485, 10528, 11242, 16065, 0, 0, 0, 0, 0, 2240, 58149, 92064, 84762, 125665, 0, 0, 0, 0, 0, 0, 228221, 760725, 805530, 722250, 1112074

Offset: 0

Peter Luschny, Nov 19 2020

The size of the centralizer of a partition p is aut(p) = Product_{j = 1..k} m(j)!*j^m(j), where m(j) is the multiplicity of j as a part of p. (For instance p = [2, 2, 2] -> aut(p) = 3!*2^3.)
Let M be the matrix with M(k, r) = Sum_{p in P(n, k)} n! / aut(p) where P(n, k) are the partitions of n with largest part k and length(p) = r. Then T(n, k) = Sum_{j=0..k} M(j, k-j+1), which are the antidiagonal sums of the upper triangular part of the matrix M.
In the example section below it is explained how the matrix M leads to a two-dimensional classification of the permutations of [n] which project to the unsigned Stirling cycle numbers and the number of permutations with longest cycle length.

			Triangle starts:
0:  [1]
1:  [0, 1]
2:  [0, 0, 2]
3:  [0, 0, 0, 6]
4:  [0, 0, 0, 3,  21]
5:  [0, 0, 0, 0,  35,   85]
6:  [0, 0, 0, 0,  55,  255,     410]
7:  [0, 0, 0, 0,   0, 1015,    1659,  2366]
8:  [0, 0, 0, 0,   0, 2485,   10528, 11242, 16065]
9:  [0, 0, 0, 0,   0, 2240,   58149, 92064, 84762, 125665]
----------------------------------------------------------
Sum  1, 1, 2, 9, 111, 6080, 2331767, ...
.
Examples for the basic two-dimensional classification of permutations (dots indicate zeros):
.
* Case n = 6:
   |   1     2     3    4    5    6  | Sum
-------------------------------------|----
1  |   .     .     .    .    .   [1] |   1
2  |   .     .   [ 15] [45] [15]     |  75
3  |   .   [ 40] [120] [40]          | 200
4  |   .   [ 90] [ 90]               | 180
5  |   .   [144]                     | 144
6  | [120]                           | 120
-------------------------------------|----
Sum| 120,  274,   225,  85,  15,  1  | 720
.
Antidiagonals: [40 + 15, 90 + 120 + 45, 120 + 144 + 90 + 40 + 15 + 1]
Leads to row 6 (disregarding leading zeros): 55 + 255 + 410 = 720.
.
* Case n = 7:
   |  1      2     3     4     5    6    7  | Sum
--------------------------------------------|-----
1  |  .      .     .     .     .    .   [1] |    1
2  |  .      .     .   [105] [105] [21]     |  231
3  |  .      .   [490] [420] [ 70]          |  980
4  |  .    [420] [630] [210]                | 1260
5  |  .    [504] [504]                      | 1008
6  |  .    [840]                            |  840
7  | [720]                                  |  720
--------------------------------------------|-----
Sum| 720,  1764,  1624, 735,  175,  21,  1  | 5040
.
Antidiagonals: [420+490+105, 504+630+420+105, 720+840+504+210+70+21+1]
Leads to row 7 (disregarding leading zeros): 1015 + 1659 + 2366 = 5040
.
* Column sums of the matrix give the unsigned Stirling cycle numbers, A132393.
* Row sums of the matrix give the number of permutations of n elements whose longest cycle have length k, A126074.
* The main antidiagonal of the matrix gives the number of n-permutations that are pure cycles of length n - k, A092271.
* The entries of the matrix sum to n!. In particular the sum over all row sums, the sum over all column sums, and the sum over all antidiagonal sums is n!.
* The columns of the triangle are finite in the sense that their entries become ultimately zero. Column sums of the triangle are A339015.

Cf. A000142 (row sums), A339015 (column sums), A132393, A126074, A092271, A121726, A339033, A006231, A002104.

# For illustration computes also A132393 and A126074 (remove the #).
def A339016Row(n):
    f = factorial(n); M = matrix(n + 2)
    for k in (0..n):
        for p in Partitions(n, max_part=k, inner=[k]):
            M[k, len(p)] += (f // p.aut())
    # print("max cyc len", [sum(M[k, j] for j in (0..n+1)) for k in (0..n)])
    # print("Stirling 1 ", [sum(M[j, k] for j in (0..n+1)) for k in (0..n)])
    if n == 0: return [1]
    return [sum(M[j, k-j+1] for j in srange(k, 0, -1)) for k in (0..n)]
for n in (0..9): print(A339016Row(n))

T(n, n) = A006231(n) + 1 = A002104(n) - (n-1) (after Franklin T. Adams-Watters in A121726).

A370948 Triangle read by rows: T(n,k) is the number of labeled forests of rooted Greg hypertrees with n white vertices and weight k, 0 <= k < n.

Original entry on oeis.org

1, 3, 1, 22, 15, 1, 262, 271, 53, 1, 4336, 6020, 2085, 165, 1, 91984, 160336, 81310, 13040, 487, 1, 2381408, 4996572, 3364011, 851690, 73024, 1407, 1, 72800928, 178613156, 150499951, 53119521, 7696794, 383649, 4041, 1

Offset: 1

Paul Laubie, Mar 06 2024

A rooted Greg hypertree is a hypertree with black and white vertices such that white vertices are labeled, black vertices are unlabeled, and each black vertex has at least two children.

The weight of a forest of rooted Greg hypertrees is the number of hypertrees minus 1 plus the weight of each hyperedge which is the number of vertices it connects minus 2. See A364709 for the analog sequence for hypertrees. A forest of rooted Greg hypertrees of weight 0 is exactly a Greg tree.

			Triangle T(n,k) begins:
n\k    0     1     2     3     4 ...
1      1;
2      3,    1;
3     22,   15,    1;
4    262,  271,   53,    1;
5   4336, 6020, 2085,  165,    1;
...

Paul Laubie, Hypertrees and embedding of the FMan operad, arXiv:2401.17439 [math.QA], 2024.

Cf. A364709, A005264 (k=0), A370949.
Row sums are A364816.
Series reversion as e.g.f. is related to A092271.

T(n)={my(x='x+O('x^(n+1))); [Vecrev(p) | p <- Vec(serlaplace(serreverse( (log(1+y*x)/y - exp(x) + x + 1)*exp(-x) )))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Mar 06 2024

E.g.f: series reversion in t of (log(1+v*t)/v - exp(t) + t + 1)*exp(-t), where the formal variable v encodes the weight.
T(n,0) = A005264(n).
T(n,n-1) = 1.

cp's OEIS Frontend

A008290 Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).

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A238363 Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.

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A339033 Triangle read by rows, T(n, k) for 0 <= k <= n. T(n, 0) = 0^n; T(n, n) = n!; otherwise T(n, k) = (n + 1 - k)*(k - 1)!.

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A211603 Triangular array read by rows: T(n,k) is the number of n-permutations that are pure cycles having exactly k fixed points; n>=2, 0<=k<=n-2.

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A121726 Sum sequence A000522 then subtract 0,1,2,3,4,5,...

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A339016 A classification of permutations based on their cycle length and the size of the centralizer of their cycle type. Triangle read by rows, T(n, k) for 0 <= k <= n.

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A370948 Triangle read by rows: T(n,k) is the number of labeled forests of rooted Greg hypertrees with n white vertices and weight k, 0 <= k < n.

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