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Search: id:A060475
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| A060475 |
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Triangular array formed from successive differences of factorial numbers, then with factorials removed. |
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+0
4
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| 1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 3, 7, 11, 9, 1, 4, 13, 32, 53, 44, 1, 5, 21, 71, 181, 309, 265, 1, 6, 31, 134, 465, 1214, 2119, 1854, 1, 7, 43, 227, 1001, 3539, 9403, 16687, 14833, 1, 8, 57, 356, 1909, 8544, 30637, 82508, 148329, 133496, 1, 9, 73, 527, 3333, 18089
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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t(n, k) is also the number of partial bijections (of an n-element set) with a fixed domain of size k and without fixed points. Equivalently, T(n, k) is the number of partial derangements with a fixed domain of size k in the symmetric inverse semigroup (monoid), I sub n. [From A. Umar (aumarh(AT)squ.edu.om), Sep 14 2008]
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REFERENCES
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Laradji, A. and Umar, A. Combinatorial results for the symmetric inverse semigroup. Semigroup Forum 75 (2007), 221-236. [From A. Umar (aumarh(AT)squ.edu.om), Sep 14 2008]
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FORMULA
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t(n, k) =A047920(n, k)/(n-k)! =(n-1)*t(n-1, k-1)+(k-1)*t(n-2, k-2) =(n-k+1)*t(n, k-1)-t(n-1, k-1)
t(n,k)=k!*sum(j=0,k,C(n-j,k-j)(-1^j)/j! [From A. Umar (aumarh(AT)squ.edu.om), Sep 14 2008]
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CROSSREFS
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Columns in one direction include A000012, A001477, A002061. Columns in other direction include A000166, A000255, A000153, A000261, A001909, A001910. Main diagonal is A002119.
Similar to A076731.
C(n, k)t(n, k)=A144089(n, k) [From A. Umar (aumarh(AT)squ.edu.om), Sep 14 2008]
Sequence in context: A102288 A107357 A026105 this_sequence A168069 A106559 A106377
Adjacent sequences: A060472 A060473 A060474 this_sequence A060476 A060477 A060478
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KEYWORD
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nonn,tabl
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Mar 16 2001
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