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%I #372 Apr 02 2024 11:44:30
%S 1,-1,1,2,-3,1,-6,11,-6,1,24,-50,35,-10,1,-120,274,-225,85,-15,1,720,
%T -1764,1624,-735,175,-21,1,-5040,13068,-13132,6769,-1960,322,-28,1,
%U 40320,-109584,118124,-67284,22449,-4536,546,-36,1,-362880,1026576,-1172700,723680,-269325,63273,-9450,870,-45,1
%N Triangle read by rows of Stirling numbers of first kind, s(n,k), n >= 1, 1 <= k <= n.
%C The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.
%C The unsigned numbers (read from right to left) also give the number of permutations of 1..n with complexity k, where the complexity of a permutation is defined to be the sum of the lengths of the cycles minus the number of cycles. In other words, the complexity equals the sum of (length of cycle)-1 over all cycles. For n=5, the numbers of permutations with complexity 0,1,2,3,4 are 1, 10, 35, 50, 24. - _N. J. A. Sloane_, Feb 08 2019
%C The unsigned numbers are also the number of permutations of 1..n with k left to right maxima (see Khovanova and Lewis, Smith).
%C With P(n) = the number of integer partitions of n, T(i,n) = the number of parts of the i-th partition of n, D(i,n) = the number of different parts of the i-th partition of n, p(j,i,n) = the j-th part of the i-th partition of n, m(j,i,n) = multiplicity of the j-th part of the i-th partition of n, Sum_[T(i,n)=k]_{i=1}^{P(n)} = sum running from i=1 to i=p(n) but taking only partitions with T(i,n)=k parts into account, Product_{j=1..T(i,n)} = product running from j=1 to j=T(i,n), Product_{j=1..D(i,n)} = product running from j=1 to j=D(i,n) one has S1(n,k) = Sum_[T(i,n)=k]_{i=1}^{P(n)} (n!/Product_{j=1..T(i,n)} p(j,i,n))* (1/Product_{j=1..D(i,n)} m(j,i,n)!). For example, S1(6,3) = 225 because n=6 has the following partitions with k=3 parts: (114), (123), (222). Their complexions are: (114): (6!/1*1*4)*(1/2!*1!) = 90, (123): (6!/1*2*3)*(1/1!*1!*1!) = 120, (222): (6!/2*2*2)*(1/3!) = 15. The sum of the complexions is 90+120+15 = 225 = S1(6,3). - _Thomas Wieder_, Aug 04 2005
%C Row sums equal 0. - _Jon Perry_, Nov 14 2005
%C |s(n,k)| enumerates unordered n-vertex forests composed of k increasing non-plane (unordered) trees. Proof from the e.g.f. of the first column and the F. Bergeron et al. reference, especially Table 1, last row (non-plane "recursive"), given in A049029. - _Wolfdieter Lang_, Oct 12 2007
%C |s(n,k)| enumerates unordered increasing n-vertex k-forests composed of k unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j >= 0 come in j+1 colors (j=0 for the k roots). - _Wolfdieter Lang_, Oct 12 2007, Feb 22 2008
%C A refinement of the unsigned array is A036039. For an association to forests of "naturally grown" rooted non-planar trees, dispositions of flags on flagpoles, and colorings of the vertices of the complete graphs K_n, see A130534. - _Tom Copeland_, Mar 30 and Apr 05 2014
%C The Stirling numbers of the first kind were related to the falling factorial and the convolved, or generalized, Bernoulli numbers B_n by Norlund in 1924 through Sum_{k=1..n+1} T(n+1,k) * x^(k-1) = (x-1)!/(x-1-n)! = (x + B.(0))^n = B_n(x), umbrally evaluated with (B.(0))^k = B_k(0) and the associated Appell polynomial B_n(x) defined by the e.g.f. (t/(exp(t) - 1))^(n+1) * exp(x*t) = exp(B.(x)t). - _Tom Copeland_, Sep 29 2015
%C With x = e^z, D_x = d/dx, D_z = d/dz, and p_n(x) the row polynomials of this entry, x^n (D_x)^n = p_n(D_z) = (D_z)! / (D_z - n)! = (xD_x)! / (xD_x - n)!. - _Tom Copeland_, Nov 27 2015
%C From the operator relation z + Psi(1) + sum_{n > 0} (-1)^n (-1/n) binomial(D,n) = z + Psi(1+D) with D = d/dz and Psi the digamma function, Zeta(n+1) = Sum_{k > n-1} (1/k) |S(k,n)| / k! for n > 0 and Zeta the Riemann zeta function. - _Tom Copeland_, Aug 12 2016
%C Let X_1,...,X_n be i.i.d. random variables with exponential distribution having mean = 1. Let Y = max{X_1,...,X_n}. Then (-1)^n*n!/( Sum_{k=1..n+1} a(n+1,k) t^(k-1) ) is the moment generating function of Y. The expectation of Y is the n-th harmonic number. - _Geoffrey Critzer_, Dec 25 2018
%C In the Ewens sampling theory describing the multivariate probability distribution of the sizes of the allelic classes in a sample of size n under the Infinite Alleles Model, |s(n,k)| gives the coefficient in the formula for the probability that a sample of n alleles has exactly k distinct types. - _Noah A Rosenberg_, Feb 10 2019
%C Named by Nielson (1906) after the Scottish mathematician James Stirling (1692-1770). - _Amiram Eldar_, Jun 11 2021 and Oct 02 2023
%C The first few row polynomials along with a recursion formula are found in a manuscript by Newton written in 1664 or 1665 (p. 169 of Turnbull) giving a geometric presentation of the binomial theorem for rational powers. - _Tom Copeland_, Dec 10 2022
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%H T. D. Noe, <a href="/A008275/b008275.txt">Rows 1 to 100 of triangle, flattened.</a>
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%H Bill Gosper, <a href="/A008275/a008275.png">Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7</a>.
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%H Niels Nörlund, <a href="https://web.archive.org/web/20160324092911/http://tocs.ulb.tu-darmstadt.de/60059613.pdf">Vorlesungen über Differenzenrechnung</a>, Chelsea Pub. Co., New York, 1954. [archive.org]
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%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Stirling Number of the First Kind</a>.
%H Thomas Wieder, <a href="/A008275/a008275.txt">Comments on A008275</a>.
%H OEIS Wiki, <a href="/wiki/Factorial_polynomials">Factorial polynomials</a>.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F s(n, k) = s(n-1, k-1) - (n-1)*s(n-1, k), n, k >= 1; s(n, 0) = s(0, k) = 0; s(0, 0) = 1.
%F The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k) = a(n-1, k-1) + (n-1)*a(n-1, k), n, k >= 1; a(n, 0) = a(0, k) = 0; a(0, 0) = 1.
%F E.g.f.: for m-th column (unsigned): ((-log(1-x))^m)/m!.
%F s(n, k) = T(n-1, k-1), n>1 and k>1, where T(n, k) is the triangle [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] and DELTA is Deléham's operator defined in A084938. The unsigned numbers are also |s(n, k)| = T(n-1, k-1), for n>0 and k>0, where T(n, k) = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...].
%F Sum_{i=0..n} (-1)^(n-i) * StirlingS1(n, i) * binomial(i, k) = (-1)^(n-k) * StirlingS1(n+1, k+1). - Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007
%F G.f.: S(n) = Product_{j=1..n} (x-j) (i.e., (x-1)*(x-2)*(x-3) = x^3 - 6*x^2 + 11*x - 6). - _Jon Perry_, Nov 14 2005
%F T(n,k) = A048993(n,k), for k=1..n. - _Reinhard Zumkeller_, Mar 18 2013
%F As lower triangular matrices A008277*A008275 = I, the identity matrix. - _Tom Copeland_, Apr 25 2014
%F a(n,k) = s(n,k) = lim_{y -> 0} Sum_{j=0..k} (-1)^j*binomial(k,j)*((-j*y)!/(-j*y-n)!)*y^(-k)/k! = Sum_{j=0..k} (-1)^(n-j)*binomial(k,j)*((j*y - 1 + n)!/(j*y-1)!)*y^(-k)/k!. - _Tom Copeland_, Aug 28 2015
%F From _Daniel Forgues_ Jan 16 2016: (Start)
%F Let x_(0) := 1 (empty product), and for n >= 1:
%F x_(n) := Product_{k=0..n-1} (x-k), called a factorial term (Boole, 1970) or a factorial polynomial (Elaydi, 2005: p. 60), and also x_(-n) := 1 / [Product_{k=0..n-1} (x+k)].
%F Then, for n >= 1: x_(n) = Sum_{k=1..n} T(n,k) * x^k, 1 / [x_(-n)] = Sum_{k=1..n} |T(n,k)| * x^k, x^n = Sum_{k=1..n} A008277(n,k) * x_(k), where A008277(n,k) are Stirling numbers of the second kind.
%F The row sums (of either signed or absolute values) are Sum_{k=1..n} T(n,k) = 0^(n-1), Sum_{k=1..n} |T(n,k)| = T(n+1,1) = n!. (End)
%F s(n,m) = ((-1)^(n-m)/n)*Sum_{i=0..m-1} C(2*n-m-i, m-i-1)*A008517(n-m+1,n-m-i+1). - _Vladimir Kruchinin_, Feb 14 2018
%F Orthogonal relation: Sum_{i=0..n} i^p*Sum_{j=k..n} (-1)^(i+j) * binomial(j,i) * Stirling1(j,k)/j! = delta(p,k), i,k,p <= n, n >= 1. - _Leonid Bedratyuk_, Jul 27 2020
%F From _Zizheng Fang_, Dec 28 2020: (Start)
%F Sum_{k=1..n} (-1)^k * k * T(n, k) = -T(n+1, 2).
%F Sum_{k=1..n} k * T(n, k) = (-1)^n * (n-2)! = T(n-1, 1) for n>=2. (End)
%F n-th row polynomial = n!*Sum_{k = 0..2*n} (-1)^(n+k)*binomial(x, k)*binomial(x-1, 2*n-k) = n!*Sum_{k = 0..2*n+1} (-1)^(n+k+1)*binomial(x, k)*binomial(x-1, 2*n+1-k). - _Peter Bala_, Mar 29 2024
%e |s(3,2)| = 3 for the three unordered 2-forest with 3 vertices and two increasing (nonplane) trees: ((1),(2,3)), ((2),(1,3)), ((3),(1,2)).
%e Triangle begins:
%e 1
%e -1, 1
%e 2, -3, 1
%e -6, 11, -6, 1
%e 24, -50, 35, -10, 1
%e -120, 274, -225, 85, -15, 1
%e 720, -1764, 1624, -735, 175, -21, 1
%e -5040, 13068, -13132, 6769, -1960, 322, -28, 1
%e 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1
%e Another version of the same triangle, from _Joerg Arndt_, Oct 05 2009: (Start)
%e s(n,k) := number of permutations of n elements with exactly k cycles ("Stirling cycle numbers")
%e n| total m=1 2 3 4 5 6 7 8 9
%e -+-----------------------------------------------------
%e 1| 1 1
%e 2| 2 1 1
%e 3| 6 2 3 1
%e 4| 24 6 11 6 1
%e 5| 120 24 50 35 10 1
%e 6| 720 120 274 225 85 15 1
%e 7| 5040 720 1764 1624 735 175 21 1
%e 8| 40320 5040 13068 13132 6769 1960 322 28 1
%e 9| 362880 40320 109584 118124 67284 22449 4536 546 36 1
%e (End)
%e |s(4,2)| = 11 for the eleven unordered 2-forest with 4 vertices, composed of two increasing (nonplane) trees: ((1),((23)(24))), ((2),((13)(14))), ((3),((12)(14))), ((4),((12)(13))); ((1),(2,3,4)),((2),(1,2,3)), ((3), (1,2,4)), ((4),(1,2,3)); ((1,2),(3,4)), ((1,3),(2,4)), ((1,4),(2,3)). - _Wolfdieter Lang_, Feb 22 2008
%p with (combinat):seq(seq(stirling1(n, k), k=1..n), n=1..10); # _Zerinvary Lajos_, Jun 03 2007
%p for i from 0 to 9 do seq(stirling1(i, j), j = 1 .. i) od; # _Zerinvary Lajos_, Nov 29 2007
%t Flatten[Table[StirlingS1[n, k], {n, 1, 10}, {k, 1, n}]] (* _Jean-François Alcover_, May 18 2011 *)
%o (PARI) T(n,k)=if(n<1,0,n!*polcoeff(binomial(x,n),k))
%o (PARI) T(n,k)=if(n<1,0,n!*polcoeff(polcoeff((1+x+x*O(x^n))^y,n),k))
%o (PARI) vecstirling(n)=Vec(factorback(vector(n-1,i,1-i*'x))) /* (A function that returns all the s(n,k) as a vector) */ \\ Bill Allombert (Bill.Allombert(AT)math.u-bordeaux1.fr), Mar 16 2009
%o (Maxima) create_list(stirling1(n+1,k+1),n,0,30,k,0,n); /* _Emanuele Munarini_, Jun 01 2012 */
%o (Haskell)
%o a008275 n k = a008275_tabl !! (n-1) !! (k-1)
%o a008275_row n = a008275_tabl !! (n-1)
%o a008275_tabl = map tail $ tail a048994_tabl
%o -- _Reinhard Zumkeller_, Mar 18 2013
%Y Diagonals: A000217, A000914, A001303, A000915, A053567, etc.
%Y Cf. A048994, A008277 (Stirling numbers of second kind), A039814, A039815, A039816, A039817, A048993, A087748.
%Y Cf. A084938, A094216, A008276 (row reversed), A008277, A008278, A094262, A121632, A130534 (unsigned version), A087755 (triangle mod 2), A000142 (row sums of absolute values).
%K sign,tabl,nice,core
%O 1,4
%A _N. J. A. Sloane_
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