A177263 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k as the last entry in the first block (1<=k<=n).
1, 0, 2, 1, 1, 4, 4, 5, 5, 10, 18, 22, 23, 23, 34, 96, 114, 118, 119, 119, 154, 600, 696, 714, 718, 719, 719, 874, 4320, 4920, 5016, 5034, 5038, 5039, 5039, 5914, 35280, 39600, 40200, 40296, 40314, 40318, 40319, 40319, 46234, 322560, 357840, 362160, 362760, 362856, 362874, 362878, 362879, 362879, 409114
Offset: 1
Examples
T(4,2)=5 because we have 12-4-3, 2-1-34, 2-1-4-3, 2-4-1-3, and 2-4-3-1 (the blocks are separated by dashes).
Triangle starts:
1;
0, 2;
1, 1, 4;
4, 5, 5, 10;
18, 22, 23, 23, 34;
96, 114, 118, 119, 119, 154;
600, 696, 714, 718, 719, 719, 874;
4320, 4920, 5016, 5034, 5038, 5039, 5039, 5914;
Links
- G. C. Greubel, Rows n = 1..50 of the triangle, flattened
- A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, Series A, Vol. 99, No. 2 (2002), pp. 345-357.
Programs
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Magma
A003422:= func< n | (&+[Factorial(j): j in [0..n-1]]) >; A177263:= func< n,k | k eq n select A003422(n) else Factorial(n-1) - Factorial(n-k-1) >; [A177263(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, May 19 2024
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Maple
T := proc (n, k) if k <= n-1 then factorial(n-1)-factorial(n-k-1) elif k = n 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
A003422[n_]:= Sum[j!, {j,0,n-1}]; T[n_, k_]:= If[k==n, A003422[n], (n-1)! -(n-k-1)!]; Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, May 19 2024 *)
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SageMath
def A003422(n): return sum(factorial(j) for j in range(n)) def A177263(n,k): return A003422(n) if k==n else factorial(n-1) - factorial(n-k-1) flatten([[A177263(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, May 19 2024
Comments