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A258579
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Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with (right) waist exactly k.
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0
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1, 1, 1, 1, 2, 3, 1, 4, 7, 11, 1, 8, 17, 30, 48, 1, 16, 43, 86, 150, 241, 1, 32, 113, 258, 492, 846, 1358, 1, 64, 307, 806, 1686, 3108, 5276, 8445, 1, 128, 857, 2610, 6012, 11904, 21392, 35904, 57256, 1, 256, 2443, 8726, 22230, 47376, 90224, 158880, 263976, 419233
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OFFSET
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0,5
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REFERENCES
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F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
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LINKS
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FORMULA
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T(n,k) = Sum_{m=0..k} binomial(k-1,m-1) * (m+1)^(n-m).
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EXAMPLE
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T(3,2) = 7 because there are exactly 7 partial idempotent mappings (of a 3-chain) with right waist exactly 2, namely: (123-->222), (123-->122), (123-->121), (12-->22), (12-->12), (23-->22), (2-->2).
Triangle starts:
1;
1,1;
1,2,3;
1,4,7,11;
1,8,17,30,48;
...
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PROG
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(PARI) mybinom(x, y) = if ((x==-1) && (y==-1), 1, binomial(x, y));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(m=0, k, mybinom(k-1, m-1) * (m+1)^(n-m)), ", "); ); print(); ); } \\ Michel Marcus, Jul 15 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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