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A172106 The triangle T_2(n, m), where T_2(n, m) is the number of surjective multi-valued functions from {1, 1, 2, 3, ..., n-1} to {1, 2, 3, ..., m} by rows (n >= 1, 1 <= m <= n).

%I A172106 #18 Apr 15 2022 04:24:34
%S A172106 0,1,1,1,4,3,1,10,21,12,1,22,93,132,60,1,46,345,900,960,360,1,94,1173,
%T A172106 4980,9300,7920,2520,1,190,3801,24612,71400,103320,73080,20160,1,382,
%U A172106 11973,113652,480060,1048320,1234800,745920,181440,1,766,37065,502500,2968560,9170280,15981840,15845760,8346240,1814400
%N A172106 The triangle T_2(n, m), where T_2(n, m) is the number of surjective multi-valued functions from {1, 1, 2, 3, ..., n-1} to {1, 2, 3, ..., m} by rows (n >= 1, 1 <= m <= n).
%C A172106 T_2(1, m) = 0 by definition. T_2(n, m) also gives the number of compositions (ordered partitions) of {1, 1, 2, 3, ..., n-1} into exactly m parts.
%H A172106 G. C. Greubel, <a href="/A172106/b172106.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A172106 M. Griffiths and I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths/griffiths11.html">A generalization of Stirling Numbers of the Second Kind via a special multiset</a>, JIS 13 (2010) #10.2.5.
%F A172106 T_2(n, m) = Sum_{j=0..m} binomial(m,j)*binomial(j+1,2)*(-1)^(m-j)*j^(n-2), for n >= 2, with T(1, 1) = 0.
%F A172106 Sum_{k=1..n} T_2(n, k) = A172109(n).
%F A172106 Sum_{k=1..n} (-1)^k*T_2(n, k) = 0. - _G. C. Greubel_, Apr 13 2022
%e A172106 Triangle begins as:
%e A172106   0;
%e A172106   1,   1;
%e A172106   1,   4,     3;
%e A172106   1,  10,    21,     12;
%e A172106   1,  22,    93,    132,      60;
%e A172106   1,  46,   345,    900,     960,     360;
%e A172106   1,  94,  1173,   4980,    9300,    7920,     2520;
%e A172106   1, 190,  3801,  24612,   71400,  103320,    73080,    20160;
%e A172106   1, 382, 11973, 113652,  480060, 1048320,  1234800,   745920,  181440;
%e A172106   1, 766, 37065, 502500, 2968560, 9170280, 15981840, 15845760, 8346240, 1814400;
%e A172106   ...
%e A172106 T_2(3, 2) = 4 since there are 4 ordered partitions of {1, 1, 2} into exactly 2 parts: (1) {{1}, {1, 2}} (2) {{1, 2}, {1}} (3) {{2}, {1, 1}} (4) {{1, 1},{2}}.
%t A172106 f[r_, n_, m_]:= Sum[Binomial[m, l] Binomial[l+r-1, r] (-1)^(m-l) l^(n-r), {l,m}]; For[n = 2, n <= 10, n++, Print[Table[f[2, n, m], {m, 1, n}]]]
%o A172106 (Magma)
%o A172106 T:= func< n,k,m | n eq 1 select 0 else (&+[(-1)^(k+j)*Binomial(k,j)*Binomial(j+m-1,m)*j^(n-m): j in [1..k]]) >;
%o A172106 [T(n,k,2): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 13 2022
%o A172106 (SageMath)
%o A172106 def T(n,k,m): return sum( (-1)^(k-j)*binomial(k,j)*binomial(j+m-1,m)*j^(n-m) for j in (1..k) ) - bool(n==1)
%o A172106 flatten([[T(n,k,2) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 13 2022
%Y A172106 This is related to A019538, A172107 and A172108.
%Y A172106 Row sums give A172109.
%K A172106 nonn,tabl
%O A172106 1,5
%A A172106 _Martin Griffiths_, Jan 25 2010