A372066 -id:A372066 - cp's OEIS frontend

A099594 Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 46, 46, 16, 1, 1, 32, 146, 230, 146, 32, 1, 1, 64, 454, 1066, 1066, 454, 64, 1, 1, 128, 1394, 4718, 6902, 4718, 1394, 128, 1, 1, 256, 4246, 20266, 41506, 41506, 20266, 4246, 256, 1, 1, 512, 12866, 85310, 237686, 329462, 237686, 85310, 12866, 512, 1

Offset: 0

Ralf Stephan, Oct 27 2004

B_n^{(-k)} is the number of distinct n by k "lonesum matrices" where a matrix of entries 0 or 1 is called lonesum when it is uniquely reconstructible from its row and column sums. [Brewbaker]
B_n^{(-k)} is the cardinality of the set { sigma in S_{n+k}: -k <= i-sigma(i) <= n for all i=1,2,...,n+k }. [Launois]
T(n,k) is also the number of permutations on [n+k] in which each substring whose support belongs to {1, 2, ..., n} or {n+1, n+2, ..., n+k} is increasing. For example, with n = 2 and k = 3, the permutation 41532 does not qualify because the substring 53 has support in {n+1, n+2, ..., n+k} = {3,4,5} but is not increasing. T(2,1) = 4 counts 123, 132, 231, 312 while the permutations satisfying Launois' condition above are 123, 132, 213, 231. A bijection between these sets of permutations would be interesting. - David Callan, Jul 22 2008. (Corrected by Norman Do, Sep 01 2008)
T(n,k) is also the number of acyclic orientations of the complete bipartite graph K_{n,k}. - Vincent Pilaud, Sep 15 2020
When indexed as a triangular array, T(n,k) is the number of permutations of [n] in which 1 is in position k and the excedance entries are precisely the entries to the left of 1. See link. - David Callan, Dec 12 2021
T(n,k) is also the number of max-closed relations between an ordered n-element set and an ordered k-element set (see the paper by Jeavons and Cooper 1995). -  Don Knuth, Feb 12 2024

			Square array begins:
  1,  1,   1,    1,     1,      1, ...
  1,  2,   4,    8,    16,     32, ...
  1,  4,  14,   46,   146,    454, ...
  1,  8,  46,  230,  1066,   4718, ...
  1, 16, 146, 1066,  6902,  41506, ...
  1, 32, 454, 4718, 41506, 329462, ...
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Arvind Ayyer and Beáta Bényi, Toppling on permutations with an extra chip, arXiv:2104.13654 [math.CO], 2021. See Table 1 (a) p. 4.
Beáta Bényi, Advances in Bijective Combinatorics, Ph. D. Dissertation, Doctoral School of Mathematics and Computer Science, University of Szeged, Bolyai Institute, 2014. See Table 1.
Beáta Bényi and Peter Hajnal, Combinatorics of poly-Bernoulli numbers, arXiv:1510.05765 [math.CO], 2015.
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016.
Beata Benyi and Peter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
Beáta Bényi and Matthieu Josuat-Vergès, Combinatorial proof of an identity on Genocchi numbers, arXiv:2010.10060 [math.CO], 2020.
Beáta Bényi and Gábor V. Nagy, Bijective enumerations of Γ-free 0-1 matrices, arXiv:1707.06899 [math.CO], 2017.
Beáta Bényi and José Luis Ramírez, On q-poly-Bernoulli numbers arising from combinatorial interpretations, arXiv:1909.09949 [math.CO], 2019.
Beáta Bényi and José Luis Ramírez, Poly-Cauchy numbers - the combinatorics behind, arXiv:2105.04791 [math.CO], 2021.
Beáta Bényi and José Luis Ramírez, Poly-Cauchy Numbers of the Second Kind-the Combinatorics Behind, Enumerative Comb. Appl. (2022) Vol. 2, No. 1, Art. #S2R1.
Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, INTEGERS Vol. 8 (2008), #A02.
David Callan, Permutations whose excedance positions are those before 1
Frédéric Chapoton and Vincent Pilaud, Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra, arXiv:2201.06896 [math.CO], 2022. See p. 12.
Peter G. Jeavons and Martin C. Cooper, Tractable constraints on ordered domains, Artificial Intelligence 79 (1995), 327-339.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
Don Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (0.1).
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024. Mentions this sequence.
Stéphane Launois, Combinatorics of H-primes in quantum matrices, Journal of Algebra, Volume 309, Issue 1, 2007, Pages 139-167.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
H. A. Witek, G. Mos and C.-P. Chou, Zhang-Zhang Polynomials of Regular 3-and 4-tier Benzenoid Strips, MATCH Commun. Math. Comput. Chem. 73 (2015) 427-442.
Wikipedia, Acyclic orientation

Rows 0..9 are A000012, A000079, A027649, A027650, A027651, A283811, A283812, A283813, A284032, A284033.
Main diagonal is A048163. Another diagonal is A188634.
Antidiagonal sums are in A098830.
Cf. A019538, A210381, A266695.

A:= (n, k)-> add(Stirling2(n+1, i+1)*Stirling2(k+1, i+1)*
             i!^2, i=0..min(n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Jan 02 2016

T[n_, k_] := Sum[(-1)^(j+n)*(1+j)^k*j!*StirlingS2[n, j], {j, 0, n}]; Table[ T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 30 2016 *)

T(n,k)=sum(j=0,n,(j+1)^k*sum(i=0,j,(-1)^(n+j-i)*binomial(j,i)*(j-i)^n))

T(n,k)=sum(j=0,min(n,k), j!^2*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2)); \\ Michel Marcus, Mar 05 2017

pB(k, n) = (-1)^n * Sum[i=0..n, (-1)^i * i! * Stirling2(n, i) / (i+1)^k ].
E.g.f.: e^(x+y) / [e^x + e^y - e^(x+y)].
T(n, k) = Sum_{j=0..n} (j+1)^k*Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n. - Paul D. Hanna, Nov 04 2004
n-th row of the array = row sums of n-th power of triangle A210381. - Gary W. Adamson, Mar 21 2012

A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 29, 13, 1, 1, 19, 73, 73, 19, 1, 1, 26, 151, 266, 151, 26, 1, 1, 34, 276, 749, 749, 276, 34, 1, 1, 43, 463, 1781, 2762, 1781, 463, 43, 1, 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1

Offset: 2

Emeric Deutsch, Jan 14 2005

Row n has n - 1 terms. Row sums yield the Fine numbers (A000957).
Related to the number of certain sets of non-crossing partitions for the root system A_n (p. 11, Athanasiadis and Savvidou). - Tom Copeland, Oct 19 2014
T(n,k) is the number of permutations pi of [n-1] with k - 1 descents such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - Colin Defant, Sep 16 2018
The absolute values of the polynomials at -1 and j (cube root of 1) seem to be given by A126120 and A005043. - F. Chapoton, Nov 16 2021
Don Knuth observes that this sequence also arrises from the enumeration of restricted max-and-min-closed relations, only there it appears as an array read by antidiagonals: see the Knuth "Notes" link and A372068. Knuth also gives a formula expressing the array A372368 in terms of this array. He also reports that there is strong experimental evidence that the n-th term of row m in this array is a polynomial of degree 2*m-2 in n. - N. J. A. Sloane, May 12 2024

			T(4, 2) = 4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD and UUU*DU*DDD, where U = (1, 1), D = (1,-1) and * indicates the peaks.
Triangle starts:
   1;
   1,  1;
   1,  4,   1;
   1,  8,   8,    1;
   1, 13,  29,   13,    1;
   1, 19,  73,   73,   19,    1;
   1, 26, 151,  266,  151,   26,    1;
   1, 34, 276,  749,  749,  276,   34,   1;
   1, 43, 463, 1781, 2762, 1781,  463,  43,  1;
   1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1;
   ...
As an array (for which the rows of the preceding triangle are the antidiagonals):
   1,  1,    1,     1,      1,      1,       1,        1,        1, ...
   1,  4,    8,    13,     19,     26,      34,       43,       53, ...
   1,  8,   29,    73,    151,    276,     463,      729,     1093, ...
   1, 13,   73,   266,    749,   1781,    3758,     7253,    13061, ...
   1, 19,  151,   749,   2762,   8321,   21659,    50471,   107833, ...
   1, 26,  276,  1781,   8321,  31004,   97754,   271125,   679355, ...
   1, 34,  463,  3758,  21659,  97754,  367285,  1196665,  3478915, ...
   1, 43,  729,  7253,  50471, 271125, 1196665,  4526470, 15118415, ...
   1, 53, 1093, 13061, 107833, 679355, 3478915, 15118415, 57500480, ...
   ...

Alois P. Heinz, Rows n = 2..142, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024

Cf. A000957, A108263, A132081.

See also A099594, A272644, A372066, A372067, A372068.

T := (n, k) -> add((j/(n-j))*binomial(n-j, k-j)*binomial(n-j,k), j=0..min(k,n-k)): for n from 2 to 13 do seq(T(n, k), k = 1..n-1) od; # yields the sequence in triangular form

T[n_, k_] := Sum[(j/(n-j))*Binomial[n-j, k-j]*Binomial[n-j, k], {j, 0, Min[k, n-k]}]; Table[T[n, k], {n, 2, 13}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

T(n, k) = Sum_{j=0..min(k, n-k)} (j/(n-j)) * C(n-j, k-j) * C(n-j, k), n >= 2.
G.f.: t*z*r/(1 - t*z*r), where r = r(t, z) is the Narayana function defined by r = z*(1 + r)*(1 + t*r).
From Tom Copeland, Oct 19 2014: (Start)
With offset 0 for A108263 and offset 1 for A132081, row polynomials of this entry P(n, x) = Sum_{i} A108263(n, i)*x^i*(1 + x)^(n - 2*i) = Sum_{i} A132081(n - 2, i)*x^i*(1 + x)^(n - 2*i).
E.g., P(4, x) = 1*x*(1 + x)^(4 - 2*1) + 2*x^2*(1 + x)^(4 - 2*2) = x + 4*x^2 + x^3.
Equivalently, let Q(n, x) be the row polynomials of A108263. Then P(n, x) = (1 + x)^n * Q(n, x/(1 + x)^2).
E.g., P(4, x) = (1 + x)^4 * (x/(1 + x)^2 + 2 * (x/(1 + x)^2)^2).
See Athanasiadis and Savvidou (p. 7). (End)

A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)(-1)^(m-i)i^(n-m)*i!, for n >= 2, m = 1..n-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 73, 29, 1, 1, 61, 301, 301, 61, 1, 1, 125, 1081, 2069, 1081, 125, 1, 1, 253, 3613, 11581, 11581, 3613, 253, 1, 1, 509, 11593, 57749, 95401, 57749, 11593, 509, 1, 1, 1021, 36301, 268381, 673261, 673261, 268381, 36301, 1021, 1

Offset: 2

N. J. A. Sloane, May 07 2016

Gives number of bitriangular permutations. Could be prefixed with row 0 containing a single 1. - N. J. A. Sloane, Jan 10 2018

			Triangle begins:
n\m  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]  1;
[3]  1,      1;
[4]  1,      5,      1;
[5]  1,     13,     13,      1;
[6]  1,     29,     73,     29,      1;
[7]  1,     61,    301,    301,     61,      1;
[8]  1,    125,   1081,   2069,   1081,    125,      1;
[9]  1,    253,   3613,  11581,  11581,   3613,    253,      1;
...

Gheorghe Coserea, Rows n = 2..101, flattened
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012.
Beáta Bényi, A Bijection for the Boolean Numbers of Ferrers Graphs, Graphs and Combinatorics (2022) Vol. 38, No. 10.
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See D_{n,k}.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
D. E. Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (16.2).
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024. Mentions this sequence.
J. Riordan, Letter to N. J. A. Sloane, Dec. 1976.

Column 2 is A036563.
Largest term in each row gives A272645.
Second diagonal from the right is 2^i - 3.
Third diagonal from the right edge is A006230.
T(2n,n) gives A048144.
For row sums see A297195.
Cf. A008277, A001469, A371761.

A272644 := proc(n,m)
    add(combinat[stirling2](m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!,i=0..m) ;
end proc:
seq(seq(A272644(n,m),m=1..n-1),n=2..10) ; # R. J. Mathar, Mar 04 2018

Table[Sum[StirlingS2[m + 1, i + 1] (-1)^(m - i) i^(n - m) i!, {i, 0, m} ], {n, 11}, {m, n - 1}] /. {} -> {0} // Flatten  (* Michael De Vlieger, May 19 2016 *)

A(n,m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
concat(vector(10, n, vector(n, m, A(n+1, m))))  \\ Gheorghe Coserea, May 16 2016

T(n,m) = Sum_{i=0..m} Stirling2(m+1, i+1)*(-1)^(m-i)*i^(n-m)*i!, for n>=2, m=1..n-1, where Stirling2(n,k) is defined by A008277.

A001469(n+1) = Sum_{m=1..2*n-1} (-1)^(m-1)*T(2*n,m). - Gheorghe Coserea, May 18 2016

More terms from Gheorghe Coserea, May 16 2016

A372067 Array read by antidiagonals: T(m,n) (m >= 0, n >= 0) = number of connected row convex (CRC) constraints between an m-element set and an n-element set.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 16, 8, 1, 1, 16, 56, 56, 16, 1, 1, 32, 176, 289, 176, 32, 1, 1, 64, 512, 1231, 1231, 512, 64, 1, 1, 128, 1408, 4623, 6655, 4623, 1408, 128, 1, 1, 256, 3712, 15887, 30553, 30553, 15887, 3712, 256, 1, 1, 512, 9472, 51103, 125197, 166186, 125197, 51103, 9472, 512, 1

Offset: 0

N. J. A. Sloane, May 12 2024, based on emails from Don Knuth, May 06 2024 and May 08 2024

See the Knuth "Notes" link for much more information about these sequences. The present sequence is called "table" in Part 1 of the Notes.

			The initial antidiagonals are:
   1,
   1, 1,
   1, 2, 1,
   1, 4, 4, 1,
   1, 8, 16, 8, 1,
   1, 16, 56, 56, 16, 1,
   1, 32, 176, 289, 176, 32, 1,
   1, 64, 512, 1231, 1231, 512, 64, 1,
   1, 128, 1408, 4623, 6655, 4623, 1408, 128, 1,
   1, 256, 3712, 15887, 30553, 30553, 15887, 3712, 256, 1,
   1, 512, 9472, 51103, 125197, 166186, 125197, 51103, 9472, 512, 1,
   ...
The array begins:
   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
   1, 4, 16, 56, 176, 512, 1408, 3712, 9472, 23552, ...
   1, 8, 56, 289, 1231, 4623, 15887, 51103, 156159, 457983, ...
   1, 16, 176, 1231, 6655, 30553, 125197, 471581, 1664061, 5572733, ...
   1, 32, 512, 4623, 30553, 166186, 790250, 3402874, 13570090, 50887322, ...
   1, 64, 1408, 15887, 125197, 790250, 4283086, 20750168, 92177312, 382005370, ...
   1, 128, 3712, 51103, 471581, 3402874, 20750168, 111803585,  547505091, 2483709151, ...
   1, 256, 9472, 156159, 1664061, 13570090, 92177312, 547505091, 2932069965, 14453287777, ...
   1, 512, 23552, 457983, 5572733, 50887322, 382005370, 2483709151, 14453287777, 76964939964, ...
   ...

Yves Deville, Olivier Barette, Pascal Van Hentenryck, Constraint satisfaction over connected row-convex constraints, Artificial Intelligence 109 (1999), 243-271.
Peter Jeavons, David Cohen, Martin C. Cooper, Constraints, consistency and closure". Artificial Intelligence 101 (1998), 251-265.

D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024

Cf. A100754, A372066, A372068.

Knuth gives a formula expressing the current array in terms of the array A372066.

A372068 Array read by antidiagonals: T(m,n) (m >= 0, n >= 0) = number of max-and-min-closed constraints between an m-element set and an n-element set.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 13, 8, 1, 1, 16, 38, 38, 16, 1, 1, 32, 104, 147, 104, 32, 1, 1, 64, 272, 506, 506, 272, 64, 1, 1, 128, 688, 1612, 2103, 1612, 688, 128, 1, 1, 256, 1696, 4856, 7887, 7887, 4856, 1696, 256, 1, 1, 512, 4096, 14016, 27477, 34088, 27477, 14016, 4096, 512, 1

Offset: 0

N. J. A. Sloane, May 12 2024, based on emails from Don Knuth, May 06 2024 and May 08 2024

See the Knuth "Notes" link for much more information about these sequences. The present sequence is called "tab" in Part 2 of the Notes.

			The initial antidiagonals are:
   1,
   1, 1,
   1, 2, 1,
   1, 4, 4, 1,
   1, 8, 13, 8, 1,
   1, 16, 38, 38, 16, 1,
   1, 32, 104, 147, 104, 32, 1,
   1, 64, 272, 506, 506, 272, 64, 1,
   1, 128, 688, 1612, 2103, 1612, 688, 128, 1,
   1, 256, 1696, 4856, 7887, 7887, 4856, 1696, 256, 1,
   1, 512, 4096, 14016, 27477, 34088, 27477, 14016, 4096, 512, 1,
   ...
The array begins:
   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
   1, 4, 13, 38, 104, 272, 688, 1696, 4096, 9728, ...
   1, 8, 38, 147, 506, 1612, 4856, 14016, 39104, 106112, ...
   1, 16, 104, 506, 2103, 7887, 27477, 90498, 285072, 865856, ...
   1, 32, 272, 1612, 7887, 34088, 134825, 498465, 1746830, 5859404, ...
   1, 64, 688, 4856, 27477, 134825, 597539, 2451038, 9455182, 34687916, ...
   1, 128, 1696, 14016, 90498, 498465, 2451038, 11055950, 46570858, 185484836, ...
   1, 256, 4096, 39104, 285072, 1746830, 9455182, 46570858, 212833803, 914854829, ...
   1, 512, 9728, 106112, 865856, 5859404, 34687916, 185484836, 914854829, 4223468802, ...
   ...

D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024

Cf. A099594, A100754, A272644, A372066, A372067.

cp's OEIS Frontend

A099594 Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).

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A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

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A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!, for n >= 2, m = 1..n-1.

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A372067 Array read by antidiagonals: T(m,n) (m >= 0, n >= 0) = number of connected row convex (CRC) constraints between an m-element set and an n-element set.

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A372068 Array read by antidiagonals: T(m,n) (m >= 0, n >= 0) = number of max-and-min-closed constraints between an m-element set and an n-element set.

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A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)(-1)^(m-i)i^(n-m)*i!, for n >= 2, m = 1..n-1.