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A156992
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Triangle t(n,k)=n!*binomial(n-1,k-1) read by rows, 1<=k<=n.
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3
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| This sequence is the ordered occupancy with no cell empty form from Riordan.
Partition {1,2,...,n} into m subsets, arrange (linearly order) the elements within each subset, then arrange the subsets. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 05 2010]
Comment 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 number of compositions of n with k summands is C(n-1,k-1), we obtain t(n,k)=n!C(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 not fixed points, that is, f(x) does not equal x for all 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. - DELEHAM Philippe, Nov 29 2011
t(n,j+k)=sum(C(n,i)*t(i,j)*t(n-i,k),i=j..n-k). - Dennis P. Walsh, Nov 29 2011
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REFERENCES
| J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 98
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FORMULA
| E.g.f. for column k is (x/(1-x))^k [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 05 2010]
t(n,k)=A000142(n)* A007318(n-1,k-1). - Dennis P. Walsh Nov 26 2011
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EXAMPLE
| 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 <shelf1><shelf2>:
<b1><b2,b3>, <b1><b3,b2>, <b2><b1,b3>, <b2><b3,b1>, <b3><b1,b2>, <b3><b2,b1> <b2,b3><b1>, <b3,b2><b1>, <b3,b1><b2>, <b3,b1><b2>, <b1,b2><b3>, <2,b1><b3>. (End)
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;
362880, 2903040, 10160640, 20321280, 25401600, 20321280, 10160640, 2903040, 362880; {3628800, 32659200, 130636800, 304819200, 457228800, 457228800, 304819200, 130636800, 32659200, 3628800;
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MAPLE
| seq(seq(n!*binomial(n-1, k-1), k=1..n), n=1..10); # Dennis P. Walsh, Nov 26 2011
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MATHEMATICA
| Clear[t, n, m];
t[n_, m_] = n!*Binomial[n - 1, m - 1];
Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}];
Flatten[%]
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CROSSREFS
| 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 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 05 2010]
Sequence in context: A192933 A079005 A178802 * A054481 A157285 A173392
Adjacent sequences: A156989 A156990 A156991 * A156993 A156994 A156995
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KEYWORD
| nonn,tabl
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 20 2009
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