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 A156992 Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows. 6
 1, 2, 2, 6, 12, 6, 24, 72, 72, 24, 120, 480, 720, 480, 120, 720, 3600, 7200, 7200, 3600, 720, 5040, 30240, 75600, 100800, 75600, 30240, 5040, 40320, 282240, 846720, 1411200, 1411200, 846720, 282240, 40320, 362880, 2903040, 10160640, 20321280, 25401600, 20321280, 10160640, 2903040, 362880 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Partition {1,2,...,n} into m subsets, arrange (linearly order) the elements within each subset, then arrange the subsets. - Geoffrey Critzer, Mar 05 2010 From Dennis P. Walsh, Nov 26 2011: (Start) Number of ways to arrange n different books in a k-shelf bookcase leaving no shelf empty. There are n! ways to arrange the books in one long line. With ni denoting the number of books for shelf i, we have n = n1 + n2 + ... + nk. Since the number of compositions of n with k summands is binomial(n-1,k-1), we obtain T(n,k) = n!*binomial(n-1,k-1) for the number of ways to arrange the n books on the k shelves. Equivalently, T(n,k) is the number of ways to stack n different alphabet blocks into k labeled stacks. Also, T(n,k) is the number of injective functions f:[n]->[n+k] such that (i) the pre-image of (n+j) exists for j=1..k and (ii) f has no fixed points, that is, for all x, f(x) does not equal x. T(n,k) is the number of labeled, rooted forests that have (i) exactly k roots, (ii) each root labeled larger than any nonroot, (iii) each root with exactly one child node, (iv) n non-root nodes, and (v) at most one child node for each node in the forest. (End) Essentially, the triangle given by (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 29 2011 T(n,j+k) = Sum_{i=j..n-k} binomial(n,i)*T(i,j)*T(n-i,k). - Dennis P. Walsh, Nov 29 2011 REFERENCES J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 98 LINKS G. C. Greubel, Rows n = 1..50 of the triangle, flattened T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy] OEIS Wiki, Sorting numbers FORMULA E.g.f. for column k is (x/(1-x))^k. - Geoffrey Critzer, Mar 05 2010 T(n,k) = A000142(n)*A007318(n-1,k-1). - Dennis P. Walsh, Nov 26 2011 Coefficient triangle of the polynomials p(n,x) = (n+1)!*hypergeom([-n],[],-x). - Peter Luschny, Apr 08 2015 EXAMPLE The triangle starts: 1; 2, 2; 6, 12, 6; 24, 72, 72, 24; 120, 480, 720, 480, 120; 720, 3600, 7200, 7200, 3600, 720; 5040, 30240, 75600, 100800, 75600, 30240, 5040; 40320, 282240, 846720, 1411200, 1411200, 846720, 282240, 40320; From Dennis P. Walsh, Nov 26 2011: (Start) T(3,2) = 12 since there are 12 ways to arrange books b1, b2, and b3 on shelves : , , , , , , , , , , , . (End) MAPLE seq(seq(n!*binomial(n-1, k-1), k=1..n), n=1..10); # Dennis P. Walsh, Nov 26 2011 with(PolynomialTools): p := (n, x) -> (n+1)!*hypergeom([-n], [], -x); seq(CoefficientList(simplify(p(n, x)), x), n=0..5); # Peter Luschny, Apr 08 2015 MATHEMATICA Table[n!*Binomial[n-1, k-1], {n, 10}, {k, n}]//Flatten PROG (Magma) [Factorial(n)*Binomial(n-1, k-1): k in [1..n], n in [1..10]]; // G. C. Greubel, May 10 2021 (Sage) flatten([[factorial(n)*binomial(n-1, k-1) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, May 10 2021 CROSSREFS Cf. A002866 (row sums). Column 1 = A000142. Column 2 = A001286 * 2! = A062119. Column 3 = A001754 * 3!. Column 4 = A001755 * 4!. Column 5 = A001777 * 5!. Column 6 = A001778 * 6!. Column 7 = A111597 * 7!. Column 8 = A111598 * 8!. Cf. A105278. - Geoffrey Critzer, Mar 05 2010 T(2n,n) gives A123072. Sequence in context: A241669 A356546 A178802 * A285529 A305215 A219694 Adjacent sequences: A156989 A156990 A156991 * A156993 A156994 A156995 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 20 2009 STATUS approved

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Last modified December 2 07:04 EST 2023. Contains 367510 sequences. (Running on oeis4.)