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A380977 Triangle read by rows: T(n,m) (1<=m<=n) = number of surjections f:[n]->[m] with f(n) != f(j), j

%I A380977 #14 May 06 2025 09:19:31
%S A380977 1,0,2,0,2,6,0,2,18,24,0,2,42,144,120,0,2,90,600,1200,720,0,2,186,
%T A380977 2160,7800,10800,5040,0,2,378,7224,42000,100800,105840,40320,0,2,762,
%U A380977 23184,204120,756000,1340640,1128960,362880,0,2,1530,72600,932400,5004720,13335840,18627840,13063680,3628800
%N A380977 Triangle read by rows: T(n,m) (1<=m<=n) = number of surjections f:[n]->[m] with f(n) != f(j), j<n.
%C A380977 Number of n-tuples containing all elements of [m] with a unique last element.
%C A380977 Consider an urn with m balls of pairwise different colors. T(n,m) is motivated by the probability p(n,m) for exactly n draws with replacement needed to obtain all colors; p(n,m)=T(n,m)/m^n. - With m fixed and n running, p(n,m) is a probability distribution. The expected number of draws needed to obtain all colors is Sum_{j=1..m} m/j. (Expected value provided by M. Shackleford.)
%H A380977 Michael Shackleford, <a href="https://mathproblems.info/working.php">Problem 74. Free gift in the cereal box problem #2</a>, Mathproblems.info.
%F A380977 T(n,m) = m!*S2(n-1,m-1) = m!*A048993(n-1,m-1).
%F A380977 T(n,m) = m*A131689(n-1,m-1).
%F A380977 T(n,3) = A068293(n-1), n>1.
%e A380977 The triangle T(n,m) begins:
%e A380977   n\m  1 2    3     4      5       6        7        8        9      10 ...
%e A380977    1:  1
%e A380977    2:  0 2
%e A380977    3:  0 2    6
%e A380977    4:  0 2   18    24
%e A380977    5:  0 2   42   144    120
%e A380977    6:  0 2   90   600   1200     720
%e A380977    7:  0 2  186  2160   7800   10800     5040
%e A380977    8:  0 2  378  7224  42000  100800   105840    40320
%e A380977    9:  0 2  762 23184 204120  756000  1340640  1128960   362880
%e A380977   10:  0 2 1530 72600 932400 5004720 13335840 18627840 13063680 3628800
%e A380977   ...
%e A380977 T(4,3)=18 is the number of 4-sequences of draws from [3] completing the covering of [3] with the last draw; these sequences are (without brackets and commas):
%e A380977    1123 1213 1223 2113 2123 2213 1132 1312 1332
%e A380977    3112 3132 3312 2231 2321 2331 3221 3231 3321
%t A380977 Table[m! StirlingS2[n - 1, m - 1], {n, 10}, {m, n}]//Flatten
%Y A380977 Cf. A048993, A068293, A131689.
%Y A380977 Row sums give A005649(n-1) for n>=1.
%K A380977 nonn,tabl
%O A380977 1,3
%A A380977 _Manfred Boergens_, Feb 10 2025