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A130534 Triangle T(n,k), 0<=k<=n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1,k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles. 48
1, 1, 1, 2, 3, 1, 6, 11, 6, 1, 24, 50, 35, 10, 1, 120, 274, 225, 85, 15, 1, 720, 1764, 1624, 735, 175, 21, 1, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 362880, 1026576, 1172700, 723680, 269325, 63273 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This triangle is an unsigned version of the triangle of Stirling numbers of the first kind, A008275, which is the main entry for these numbers. - N. J. A. Sloane, Jan 25 2011

Or, triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,2,3,3,4,4,5,5,6,6,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

Reversal of A094638.

Equals A132393*A007318, as infinite lower triangular matrices. - Philippe Deléham, Nov 13 2007

From Johannes W. Meijer, Oct 07 2009: (Start)

The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the exponential integrals E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + n*(n+1)/x^2 - n*(n+1)*(n+2)/x^3 + .. ), see Abramowitz and Stegun. This formula follows from the general formula for the asymptotic expansion, see A163932. We rewrite E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + (n^2+n)/x^2 - (2*n+3*n^2+n^3)/x^3 + (6*n+11*n^2+6*n^3+n^4)/x^3 - .. ) and observe that the T(n,m) are the polynomials coefficients in the denominators. Looking at the a(n,m) formula of A028421, A163932 and A163934, and shifting the offset given above to 1, we can write T(n-1,m-1) = a(n,m) = (-1)^(n+m)*stirling1(n,m), see the Maple program.

The asymptotic expansion leads for values of n from one to eleven to known sequences, see the cross-references. With these sequences one can form the triangles A008279 (right hand columns) and A094587 (left hand columns).

See A163936 for information about the o.g.f.s. of the right hand columns of this triangle.

(End)

The number of elements greater than i to the left of i in a permutation gives the i-th element of the inversion vector. (Skiena-Pemmaraju 2003 p. 69.) T(n,k) is the number of n-permutations that have exactly k 0's in their inversion vector. See evidence in Mathematica code below. - Geoffrey Critzer, May 07 2010

T(n,k) counts the rooted trees with k+1 trunks in forests of "naturally grown" rooted trees with n+2 nodes. This corresponds to sums of coefficients of iterated derivatives representing vectors, Lie derivatives, or infinitesimal generators for flow fields and formal group laws. Cf. links in A139605. - Tom Copeland, Mar 23 2014

A refinement is A036039. - Tom Copeland, Mar 30 2014

From Tom Copeland, Apr 05 2014: (Start)

With initial n=1 and row polynomials of T as p(n,x)=x(x+1)...(x+n-1), the powers of x correspond to the number of trunks of the rooted trees of the "naturally-grown" forest referred to above. With each trunk allowed m colors, p(n,m) counts the number of such non-plane colored trees for the forest with each tree having n+1 vertices.

p(2,m) = m + m^2 = A002378(m) = 2*A000217(m) = 2*first subdiag of |A238363|.

p(3,m) = 2m + 3m^2 + m^3 = A007531(m+2) = 3*A007290(m+2) = 3*(second subdiag A238363).

p(4,m) = 6m + 11m^2 + 6m^3 + m^4 = A052762(m+3) = 4*A033487(m) = 4*third subdiag.

From the Joni et al. link, p(n,m) also represents the disposition of n distinguishable flags on m distinguishable flagpoles.

The chromatic polynomial for the complete graph K_n is the falling factorial, which encodes the colorings of the n vertices of K_n and gives a shifted version of p(n,m).

E.g.f. for the row polynomials: (1-y)^(-x).

(End)

A relation to derivatives of the determinant |V(n)| of the n-by-n Vandermonde matrix V(n) in the indeterminates c(1) thru c(n):

  |V(n)| = Product(1<=j<k<=n, (c(j)-c(k))). Let W(n,x) = |V(n)|*(c(1)c(2)...c(n))^x, then p(n,x) = W^(-1)[c(1)d/dc(1)...c(n)d/dc(n)]W. This is a variant of the Cayley identity. See Chervov link, p. 47. - Tom Copeland, Apr 10 2014

From Peter Bala, Jul 21 2014: (Start)

Let M denote the lower unit triangular array A094587 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

/I_k 0\

\ 0  M/

having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

For the relation of this rising factorial to the moments of Viennot's Laguerre stories, see the Hetyei link p. 4. - Tom Copeland, Oct 01 2015

Can also be seen as the Bell transform of n! without column 0 (and shifted enumeration). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

REFERENCES

Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge University Press, 2003, pp. 69-71 [From Geoffrey Critzer, May 07 2010]

LINKS

T. D. Noe, Rows n=0..50 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251. [From Johannes W. Meijer, Oct 07 2009]

A. Chervov, Decomplexification of the Capelli identities and holomorphic factorization, arxiv 1203.5759 [math.QA], Mar 2012. [Tom Copeland, Apr 10 2014]

FindStat - Combinatorial Statistic Finder, The number of saliances of the permutation, The number of cycles in the cycle decomposition of a permutation.

Martin Griffiths, Generating Functions for Extended Stirling Numbers of the First Kind, Journal of Integer Sequences, 17 (2014), #14.6.4.

G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009.

S. Joni, G. Rota, and B. Sagan, From Sets to Functions: Three Elementary Examples, Discrete Mathematics, vol. 37, no. 2-3, pp. 193-202, 1981. [Tom Copeland, Apr 05 2014]

Dennis Walsh, A short note on unsigned Stirling numbers

FORMULA

T(0,0)=1, T(n,k)=0 if k>n or if n<0, T(n,k)=T(n-1,k-1)+n*T(n-1,k). T(n,0)=n!=A000142(n). T(2*n,n)=A129505(n+1). Sum_{k, 0<=k<=n}T(n,k)=(n+1)!=A000142(n+1). Sum_{k, 0<=k<=n}T(n,k)^2=A047796(n+1). T(n,k)=|Stirling1(n+1,k+1)|, see A008275. (x+1)(x+2)...(x+n)=Sum_{k, 0<=k<=n}T(n,k)*x^k. [Corrected by Arie Bos, Jul 11 2008]

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. - Philippe Deléham, Nov 13 2007

For 1<=k<=n, let A={a1,a2,...,ak} denote a size-k subset of {1,2,...,n}. Then T(n,n-k)=sum(product(ai)) where the sum is over all subsets A and the product is over i=1,..,k. For example, T(4,1)=50 since (1)(2)(3)+(1)(2)(4)+(1)(3)(4)+(2)(3)(4)=50. - Dennis P. Walsh, Jan 25 2011

The preceding formula means T(n,k) = sigma_{n-k}(1,2,3,..,n) with the (n-k)-th elementary symmetric function sigma with the indeterminates chosen as 1,2,...,n. See the Oct 24 2011 comment in A094638 with sigma called there a. - Wolfdieter Lang, Feb 06 2013.

n-th row of the triangle = top row of M^n, where M is the production matrix:

1, 1

1, 2, 1

1, 3, 3, 1

1, 4, 6, 4, 1

...

- Gary W. Adamson, Jul 08 2011

Exponential Riordan array [1/(1 - x), log(1/(1 - x))]. Recurrence: T(n+1,k+1) = sum {i = 0..n-k} (n + 1)!/(n + 1 - i)!*T(n-i,k). - Peter Bala, Jul 21 2014

EXAMPLE

Triangle  T(n,k) begins:

n\k       0      1      2      3      4     5    6   7  8 ...

n=0:      1

n=1:      1      1

n=2:      2      3      1

n=3:      6     11      6      1

n=4:     24     50     35     10      1

n=5:    120    274    225     85     15     1

n=6:    720   1764   1624    735    175    21    1

n=7:   5040  13068  13132   6769   1960   322   28   1

n=8:  40320 109584 118124  67284  22449  4536  546  36  1

n=9: 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1;

n=10: 3628800, 10628640, 12753576, 8409500, 3416930, 902055, 157773, 18150, 1320, 55, 1.

[Reformatted and extended by Wolfdieter Lang, Feb 05 2013]

T(3,2) = 6 because there are 6 permutations of {1,2,3,4} that have exactly 2 0's in their inversion vector:{1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2}, {2, 1, 3, 4},{2, 3, 1, 4}, {2, 3, 4, 1}. The respective inversion vectors are: {0, 0, 1},{0, 1, 0}, {0, 2, 0}, {1, 0, 0}, {2, 0, 0}, {3, 0, 0}. - Geoffrey Critzer, May 07 2010

T(3,1)=11 since there are exactly 11 permutations of {1,2,3,4} with exactly 2 cycles, namely, (1)(234),(1)(243),(2)(134),(2)(143),(3)(124),(3)(142),

  (4)(123),(4)(143),(12)(34),(13)(24), and (14)(23). - Dennis P. Walsh, Jan 25 2011

From Peter Bala, Jul 21 2014: (Start)

With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins

/ 1          \/1        \/1        \      / 1           \

| 1  1       ||0 1      ||0 1      |      | 1  1        |

| 2  2  1    ||0 1 1    ||0 0 1    |... = | 2  3  1     |

| 6  6  3 1  ||0 2 2 1  ||0 0 1 1  |      | 6 11  6 1   |

|24 24 12 4 1||0 6 6 3 1||0 0 2 2 1|      |24 50 35 10 1|

|...         ||...      ||...      |      |...          |

(End)

MAPLE

with(combinat): A130534 := proc(n, m): (-1)^(n+m)*stirling1(n+1, m+1) end proc: seq(seq(A130534(n, m), m=0..n), n=0..10); # Johannes W. Meijer, Oct 07 2009, revised Sep 11 2012

# The function BellMatrix is defined in A264428.

# Adds (1, 0, 0, 0, ..) as column 0 (and shifts the enumeration).

BellMatrix(n -> n!, 9); # Peter Luschny, Jan 27 2016

MATHEMATICA

Table[Table[ Length[Select[Map[ToInversionVector, Permutations[m]], Count[ #, 0] == n &]], {n, 0, m - 1}], {m, 0, 8}] // Grid (* Geoffrey Critzer, May 07 2010 *)

PROG

(Haskell)

a130534 n k = a130534_tabl !! n !! k

a130534_row n = a130534_tabl !! n

a130534_tabl = map (map abs) a008275_tabl

-- Reinhard Zumkeller, Mar 18 2013

CROSSREFS

See A008275, which is the main entry for these numbers.

Diagonals: A000012 A000217 A000914 A001303 A000915 A053567 A112002 A191685. Columns A000142 A000254 A000399 A000454 A000482 A001233 A001234.

From Johannes W. Meijer, Oct 07 2009: (Start)

Row sums equal A000142.

The asymptotic expansions lead to A000142 (n=1), A000142(n=2; minus a(0)), A001710 (n=3), A001715 (n=4), A001720 (n=5), A001725 (n=6), A001730 (n=7), A049388 (n=8), A049389 (n=9), A049398 (n=10), A051431 (n=11), A008279 and A094587.

Cf. A163931 (E(x,m,n)), A028421 (m=2), A163932 (m=3), A163934 (m=4), A163936.

(End)

Cf. A136662.

Sequence in context: A121748 A174893 A008275 * A107416 A105613 A263634

Adjacent sequences:  A130531 A130532 A130533 * A130535 A130536 A130537

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Aug 09 2007

STATUS

approved

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Last modified July 28 08:25 EDT 2016. Contains 275132 sequences.