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A177264
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k as the first entry in the last block (1<=k<=n).
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2
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1, 2, 0, 4, 1, 1, 10, 5, 5, 4, 34, 23, 23, 22, 18, 154, 119, 119, 118, 114, 96, 874, 719, 719, 718, 714, 696, 600, 5914, 5039, 5039, 5038, 5034, 5016, 4920, 4320, 46234, 40319, 40319, 40318, 40314, 40296, 40200, 39600, 35280, 409114, 362879, 362879, 362878, 362874, 362856, 362760, 362160, 357840, 322560
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OFFSET
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1,2
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COMMENTS
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A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 45123867 has 4 blocks: 45, 123, 8, and 67.
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LINKS
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FORMULA
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T(n, k) = (n-1)! - (k-2)! if 2 <= k <= n, otherwise T(n, 1) = 0! + 1! + ... + (n-1)! = A003422(n).
T(n, k) = A177263(n, n-k+1) (mirror image).
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EXAMPLE
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T(4,3)=5 because we have 12-4-3, 2-1-34, 2-1-4-3, 2-4-1-3, and 4-2-1-3 (the blocks are separated by dashes).
Triangle starts:
1;
2, 0;
4, 1, 1;
10, 5, 5, 4;
34, 23, 23, 22, 18;
154, 119, 119, 118, 114, 96;
874, 719, 719, 718, 714, 696, 600;
5914, 5039, 5039, 5038, 5034, 5016, 4920, 4320;
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MAPLE
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T := proc (n, k) if 2 <= k and k <= n then factorial(n-1)-factorial(k-2) elif k = 1 then sum(factorial(j), j = 0 .. n-1) else 0 end if end proc: for n to 10 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form
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MATHEMATICA
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A003422[n_]:= Sum[j!, {j, 0, n-1}];
T[n_, k_]:= If[k==1, A003422[n], (n-1)! -(k-2)!];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, May 19 2024 *)
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PROG
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(Magma)
A003422:= func< n | (&+[Factorial(j): j in [0..n-1]]) >;
A177264:= func< n, k | k eq 1 select A003422(n) else Factorial(n-1) - Factorial(k-2) >;
(SageMath)
def A003422(n): return sum(factorial(j) for j in range(n))
def A177264(n, k): return A003422(n) if k==1 else factorial(n-1) - factorial(k-2)
flatten([[A177264(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, May 19 2024
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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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