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A056940 Number of antichains (or order ideals) in the poset 4*m*n or plane partitions with at most m rows and n columns and entries <= 4. 17

%I #53 May 14 2022 19:50:01

%S 1,1,1,1,5,1,1,15,15,1,1,35,105,35,1,1,70,490,490,70,1,1,126,1764,

%T 4116,1764,126,1,1,210,5292,24696,24696,5292,210,1,1,330,13860,116424,

%U 232848,116424,13860,330,1,1,495,32670,457380,1646568,1646568,457380,32670,495,1

%N Number of antichains (or order ideals) in the poset 4*m*n or plane partitions with at most m rows and n columns and entries <= 4.

%C Triangle of generalized binomial coefficients (n,k)_4; cf. A342889. - _N. J. A. Sloane_, Apr 03 2021

%C Determinants of 4 X 4 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - _Gerald McGarvey_, Feb 24 2005

%C Row sums are: {1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, ...}. - _Roger L. Bagula_, Mar 08 2010

%C Also determinants of 4x4 arrays whose entries come from a single row: T(n,k) = det [C(n,k), C(n,k-1), C(n,k-2), C(n,k-3); C(n,k+1), C(n,k), C(n,k-1), C(n,k-2); C(n,k+2), C(n,k+1), C(n,k), C(n,k-1); C(n,k+3), C(n,k+2), C(n,k+1), C(n,k)]. - _Peter Bala_, May 10 2012

%H G. C. Greubel, <a href="/A056940/b056940.txt">Rows n = 0..100 of triangle, flattened</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

%H J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]

%H Johann Cigler, <a href="https://arxiv.org/abs/2103.01652">Pascal triangle, Hoggatt matrices, and analogous constructions</a>, arXiv:2103.01652 [math.CO], 2021.

%H Johann Cigler, <a href="https://www.researchgate.net/publication/349376205_Some_observations_about_Hoggatt_triangles">Some observations about Hoggatt triangles</a>, Universität Wien (Austria, 2021).

%H P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>, sect. 495, 1916.

%H R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/12-2.pdf">Theory and application of plane partitions, II</a>. Studies in Appl. Math. 50 (1971), p. 259-279. <a href="http://doi.org/10.1002/sapm1971503259">DOI:10.1002/sapm1971503259</a>. Thm. 18.1.

%H <a href="/index/Pos#posets">Index entries for sequences related to posets</a>

%F Product_{k=0..3} C(n+m+k, m+k)/C(n+k, k) gives the array as a square.

%F T(n,m,q) = c(n,q)/(c(m,q)*c(n-m,q)) with c(n,q) = Product_{i=1..n, j=0..q} (i + j), q = 3. - _Roger L. Bagula_, Mar 08 2010

%F From _Peter Bala_, Oct 13 2011: (Start)

%F T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1).

%F Define f(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is f(r,0)*f(r,n)/(f(r,k)*f(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)

%e Triangle begins as:

%e 1.

%e 1, 1.

%e 1, 5, 1.

%e 1, 15, 15, 1.

%e 1, 35, 105, 35, 1.

%e 1, 70, 490, 490, 70, 1.

%e 1, 126, 1764, 4116, 1764, 126, 1.

%e 1, 210, 5292, 24696, 24696, 5292, 210, 1.

%e 1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1. - _Roger L. Bagula_, Mar 08 2010

%t c[n_, q_] = Product[i + j, {j, 0, q}, {i, 1, n}];

%t T[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);

%t Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* _Roger L. Bagula_, Mar 08 2010 *)(* modified by _G. C. Greubel_, Apr 13 2019 *)

%o (PARI) A056940(n,m)=prod(k=0,3,binomial(n+m+k,m+k)/binomial(n+k,k)) \\ _M. F. Hasler_, Sep 26 2018

%Y Cf. A000372, A056932, A001263.

%Y Antidiagonals sum to A005362 (Hoggatt sequence).

%Y Cf. A056939 (q=2), A056940 (q=3), A056941 (q=4), A142465 (q=5), A142467 (q=6), A142468 (q=7), A174109 (q=8).

%Y Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

%K nonn,easy,tabl

%O 0,5

%A _Mitch Harris_

%E Edited by _M. F. Hasler_, Sep 26 2018

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