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%I #203 Mar 24 2024 13:11:37
%S 1,1,1,2,3,1,6,11,6,1,24,50,35,10,1,120,274,225,85,15,1,720,1764,1624,
%T 735,175,21,1,5040,13068,13132,6769,1960,322,28,1,40320,109584,118124,
%U 67284,22449,4536,546,36,1,362880,1026576,1172700,723680,269325,63273,9450,870,45,1
%N 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.
%C 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
%C 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.
%C Reversal of A094638.
%C Equals A132393*A007318, as infinite lower triangular matrices. - _Philippe Deléham_, Nov 13 2007
%C From _Johannes W. Meijer_, Oct 07 2009: (Start)
%C 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.
%C 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).
%C See A163936 for information about the o.g.f.s. of the right-hand columns of this triangle.
%C (End)
%C 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
%C 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
%C A refinement is A036039. - _Tom Copeland_, Mar 30 2014
%C From _Tom Copeland_, Apr 05 2014: (Start)
%C 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) gives the number of such non-plane colored trees for the forest with each tree having n+1 vertices.
%C p(2,m) = m + m^2 = A002378(m) = 2*A000217(m) = 2*(first subdiag of |A238363|).
%C p(3,m) = 2m + 3m^2 + m^3 = A007531(m+2) = 3*A007290(m+2) = 3*(second subdiag A238363).
%C p(4,m) = 6m + 11m^2 + 6m^3 + m^4 = A052762(m+3) = 4*A033487(m) = 4*(third subdiag).
%C From the Joni et al. link, p(n,m) also represents the disposition of n distinguishable flags on m distinguishable flagpoles.
%C 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).
%C E.g.f. for the row polynomials: (1-y)^(-x).
%C (End)
%C A relation to derivatives of the determinant |V(n)| of the n X n Vandermonde matrix V(n) in the indeterminates c(1) thru c(n):
%C |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
%C From _Peter Bala_, Jul 21 2014: (Start)
%C 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
%C /I_k 0\
%C \ 0 M/
%C 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)
%C 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
%C 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
%D Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge University Press, 2003, pp. 69-71. [_Geoffrey Critzer_, May 07 2010]
%H T. D. Noe, <a href="/A130534/b130534.txt">Rows n = 0..50 of triangle, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251. [From _Johannes W. Meijer_, Oct 07 2009]
%H A. Chervov, <a href="http://arxiv.org/abs/1203.5759">Decomplexification of the Capelli identities and holomorphic factorization</a>, arxiv 1203.5759 [math.QA], Mar 2012. [_Tom Copeland_, Apr 10 2014]
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000007">The number of saliances of the permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000031">The number of cycles in the cycle decomposition of a permutation</a>.
%H Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Griffiths/griffiths31.html">Generating Functions for Extended Stirling Numbers of the First Kind</a>, Journal of Integer Sequences, 17 (2014), #14.6.4.
%H G. Hetyei, <a href="http://arxiv.org/abs/0909.4352">Meixner polynomials of the second kind and quantum algebras representing su(1,1)</a>, arXiv preprint arXiv:0909.4352 [math.QA], 2009.
%H S. Joni, G. Rota, and B. Sagan, <a href="http://dx.doi.org/10.1016/0012-365X(81)90219-3">From Sets to Functions: Three Elementary Examples</a>, Discrete Mathematics, vol. 37, no. 2-3, pp. 193-202, 1981. [_Tom Copeland_, Apr 05 2014]
%H Matthieu Josuat-Verges, <a href="https://arxiv.org/abs/1610.02965">A q-analog of Schläfli and Gould identities on Stirling numbers</a>, Preprint, arXiv:1610.02965 [math.CO], 2016.
%H Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H Lucas Sá and Antonio M. García-García, <a href="https://arxiv.org/abs/2104.07647">The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos</a>, arXiv:2104.07647 [hep-th], 2021.
%H Igor Victorovich Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2024-02-2.pdf#page=15">On the ordinal numbers of triangles of generalized special numbers</a>, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.
%H Dennis Walsh, <a href="https://web.archive.org/web/20111208160649/http://frank.mtsu.edu/~dwalsh/STIRLIN1.pdf">A short note on unsigned Stirling numbers</a>
%F 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..n} T(n,k) = (n+1)! = A000142(n+1). Sum_{k=0..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..n} T(n,k)*x^k. [Corrected by _Arie Bos_, Jul 11 2008]
%F Sum_{k=0..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
%F For k=1..n, let A={a_1,a_2,...,a_k} denote a size-k subset of {1,2,...,n}. Then T(n,n-k) = Sum(Product_{i=1..k} a_i) where the sum is over all subsets A. 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
%F 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
%F From _Gary W. Adamson_, Jul 08 2011: (Start)
%F n-th row of the triangle = top row of M^n, where M is the production matrix:
%F 1, 1;
%F 1, 2, 1;
%F 1, 3, 3, 1;
%F 1, 4, 6, 4, 1;
%F ... (End)
%F 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
%e Triangle T(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10
%e n=0: 1
%e n=1: 1 1
%e n=2: 2 3 1
%e n=3: 6 11 6 1
%e n=4: 24 50 35 10 1
%e n=5: 120 274 225 85 15 1
%e n=6: 720 1764 1624 735 175 21 1
%e n=7: 5040 13068 13132 6769 1960 322 28 1
%e n=8: 40320 109584 118124 67284 22449 4536 546 36 1
%e n=9: 362880 1026576 1172700 723680 269325 63273 9450 870 45 1
%e n=10: 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1
%e [Reformatted and extended by _Wolfdieter Lang_, Feb 05 2013]
%e 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
%e 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
%e From _Peter Bala_, Jul 21 2014: (Start)
%e With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
%e / 1 \/1 \/1 \ / 1 \
%e | 1 1 ||0 1 ||0 1 | | 1 1 |
%e | 2 2 1 ||0 1 1 ||0 0 1 |... = | 2 3 1 |
%e | 6 6 3 1 ||0 2 2 1 ||0 0 1 1 | | 6 11 6 1 |
%e |24 24 12 4 1||0 6 6 3 1||0 0 2 2 1| |24 50 35 10 1|
%e |... ||... ||... | |... |
%e (End)
%p 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
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0 (and shifts the enumeration).
%p BellMatrix(n -> n!, 9); # _Peter Luschny_, Jan 27 2016
%t Table[Table[ Length[Select[Map[ToInversionVector, Permutations[m]], Count[ #, 0] == n &]], {n, 0, m - 1}], {m, 0, 8}] // Grid (* _Geoffrey Critzer_, May 07 2010 *)
%t rows = 10;
%t t = Range[0, rows]!;
%t T[n_, k_] := BellY[n, k, t];
%t Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o (Haskell)
%o a130534 n k = a130534_tabl !! n !! k
%o a130534_row n = a130534_tabl !! n
%o a130534_tabl = map (map abs) a008275_tabl
%o -- _Reinhard Zumkeller_, Mar 18 2013
%Y See A008275, which is the main entry for these numbers; A094638 (reversed rows).
%Y Diagonals: A000012 A000217 A000914 A001303 A000915 A053567 A112002 A191685. Columns A000142 A000254 A000399 A000454 A000482 A001233 A001234.
%Y From _Johannes W. Meijer_, Oct 07 2009: (Start)
%Y Row sums equal A000142.
%Y 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.
%Y Cf. A163931 (E(x,m,n)), A028421 (m=2), A163932 (m=3), A163934 (m=4), A163936.
%Y (End)
%Y Cf. A136662.
%K nonn,tabl
%O 0,4
%A _Philippe Deléham_, Aug 09 2007
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