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).
0, 1, 1, 1, 4, 3, 1, 10, 21, 12, 1, 22, 93, 132, 60, 1, 46, 345, 900, 960, 360, 1, 94, 1173, 4980, 9300, 7920, 2520, 1, 190, 3801, 24612, 71400, 103320, 73080, 20160, 1, 382, 11973, 113652, 480060, 1048320, 1234800, 745920, 181440, 1, 766, 37065, 502500, 2968560, 9170280, 15981840, 15845760, 8346240, 1814400
Offset: 1
Examples
Triangle begins as:
0;
1, 1;
1, 4, 3;
1, 10, 21, 12;
1, 22, 93, 132, 60;
1, 46, 345, 900, 960, 360;
1, 94, 1173, 4980, 9300, 7920, 2520;
1, 190, 3801, 24612, 71400, 103320, 73080, 20160;
1, 382, 11973, 113652, 480060, 1048320, 1234800, 745920, 181440;
1, 766, 37065, 502500, 2968560, 9170280, 15981840, 15845760, 8346240, 1814400;
...
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}}.
Links
- G. C. Greubel, Rows n = 1..50 of the triangle, flattened
- M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
Programs
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Magma
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]]) >; [T(n,k,2): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 13 2022
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Mathematica
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}]]] -
SageMath
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) flatten([[T(n,k,2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 13 2022
Formula
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.
Sum_{k=1..n} T_2(n, k) = A172109(n).
Sum_{k=1..n} (-1)^k*T_2(n, k) = 0. - G. C. Greubel, Apr 13 2022
Comments