A011971 Aitken's array: triangle of numbers {a(n,k), n >= 0, 0 <= k <= n} read by rows, defined by a(0,0)=1, a(n,0) = a(n-1,n-1), a(n,k) = a(n,k-1) + a(n-1,k-1).

1, 1, 2, 2, 3, 5, 5, 7, 10, 15, 15, 20, 27, 37, 52, 52, 67, 87, 114, 151, 203, 203, 255, 322, 409, 523, 674, 877, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147, 21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975

Offset: 0

Author

N. J. A. Sloane, J. H. Conway and R. K. Guy

Keywords

• tabl
• nonn
• easy
• nice

Comments

Also called the Bell triangle or the Peirce triangle.

a(n,k) is the number of equivalence relations on {0, ..., n} such that k is not equivalent to n, k+1 is not equivalent to n, ..., n-1 is not equivalent to n. - Don Knuth, Sep 21 2002 [Comment revised by Thijs van Ommen (thijsvanommen(AT)gmail.com), Jul 13 2008]

Named after the New Zealand mathematician Alexander Craig Aitken (1895-1967). - Amiram Eldar, Jun 11 2021

Examples

			Triangle begins:
00:       1
01:       1      2
02:       2      3      5
03:       5      7     10     15
04:      15     20     27     37     52
05:      52     67     87    114    151    203
06:     203    255    322    409    523    674    877
07:     877   1080   1335   1657   2066   2589   3263   4140
08:    4140   5017   6097   7432   9089  11155  13744  17007  21147
09:   21147  25287  30304  36401  43833  52922  64077  77821  94828 115975
10:  115975 137122 162409 192713 229114 272947 325869 389946 467767 562595 678570
...
		

References

• Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 205.
• Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 212.
• Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418).
• Charles Sanders Peirce, On the Algebra of Logic, American Journal of Mathematics, Vol. 3, pages 15-57, 1880. Reprinted in Collected Papers (1935-1958) and in Writings of Charles S. Peirce: A Chronological Edition (Indiana University Press, Bloomington, IN, 1986).
• Jeffrey Shallit, A triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71.

Links

• T. D. Noe and Chai Wah Wu, Rows n = 0..200 of triangle, flattened (rows n = 0..50 from T. D. Noe)
• Alexander Craig Aitken, A problem in combinations, Edinburgh Mathematical Notes, Vol. 28 (1933), pp. xviii-xxiii.
• H. W. Becker, Rooks and rhymes, Math. Mag., Vol. 22, No. 1 (1948/49), pp. 23-26. See Table IV.
• H. W. Becker, Rooks and rhymes, Math. Mag., Vol. 22, No. 1 (1948/49), pp. 23-26. [Annotated scanned copy]
• Antonio Bernini, Mathilde Bouvel and Luca Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18, No. 3-4 (2007), pp. 223-237 (see array p. 228).
• Clarence H. Best, Jerry Griggs, and Ira Gessel, Partitions of finite sets, Advanced Problem 6151, The American Mathematical Monthly, Vol. 86, No. 1 (Jan., 1979), pp. 64-65.
• Robert W. Donley, Jr., Binomial arrays and generalized Vandermonde identities, arXiv:1905.01525 [math.CO], 2019.
• Martin Cohn, Shimon Even, Karl Menger, Jr. and Philip K. Hooper, On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly, Vol. 69, No. 8 (1962), pp. 782-785. MR1531841.
• Martin Cohn, Shimon Even, Karl Menger, Jr. and Philip K. Hooper, On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly Vol. 69, No. 8 (1962), pp. 782-785. MR1531841. [Annotated scanned copy]
• Dominique Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
• Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968.
• Nick Hobson, Python program for this sequence.
• Don Knuth, Email to N. J. A. Sloane, Jan 29 2018.
• Charles Sanders Peirce, Assorted Papers.
• Charles Sanders Peirce, Collected Papers.
• Charles Sanders Peirce, Published Works.
• Jeffrey Shallit, A triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71.
• Todd Tichenor, Bounds on graph compositions and the connection to the Bell triangle, Discr. Math., Vol. 339, No. 4 (2016), pp. 1419-1423.
• Eric Weisstein's World of Mathematics, Bell Triangle.
• Denys Wuilquin, Letters to N. J. A. Sloane, August 1984.

Crossrefs

Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, etc., A011968, A011969. Cf. A046934, A011972 (duplicates removed).

Main diagonal is in A094577. Mirror image is in A123346.

See also A095149, A106436, A108041, A108042, A108043.

Programs

GAP

T:=Flat(List([0..9],n->List([0..n],k->Sum([0..k],i->Binomial(k,i)*Bell(n-k+i))))); # Muniru A Asiru, Oct 26 2018

Haskell

a011971 n k = a011971_tabl !! n !! k
a011971_row n = a011971_tabl !! n
a011971_tabl = iterate (\row -> scanl (+) (last row) row) [1]
-- Reinhard Zumkeller, Dec 09 2012

Maple

A011971 := proc(n,k) option remember; if n=0 and k=0 then 1 elif k=0 then A011971(n-1,n-1) else A011971(n,k-1)+A011971(n-1,k-1); fi: end;
for n from 0 to 12 do lprint([ seq(A011971(n,k),k=0..n) ]); od:
# Compare the analogue algorithm for the Catalan numbers in A030237.
BellTriangle := proc(len) local P, T, n; P := [1]; T := [[1]];
for n from 1 to len - 1 do P := ListTools:-PartialSums([P[-1], op(P)]);
T := [op(T), P] od; T end:
BellTriangle(6); ListTools:-Flatten(%); # Peter Luschny, Mar 26 2022

Mathematica

a[0, 0] = 1; a[n_, 0] := a[n, 0] = a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Flatten[ Table[ a[n, k], {n, 0, 9}, {k, 0, n}]] (* Robert G. Wilson v, Mar 27 2004 *)
Flatten[Table[Sum[Binomial[k, i]*BellB[n-k+i], {i, 0, k}], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, May 24 2016, after Vladeta Jovovic *)

Python

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011971 = blist = [1]
for _ in range(10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A011971 += blist # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 19 2014

Formula

Double-exponential generating function: Sum_{n, k} a(n-k, k) x^n y^k / n! k! = exp(e^{x+y}-1+x). - Don Knuth, Sep 21 2002 [U coordinates, reversed]

a(n,k) = Sum_{i=0..k} binomial(k,i)*Bell(n-k+i). - Vladeta Jovovic, Oct 15 2006

Extensions

Peirce reference from Jon Awbrey, Mar 11 2002

Reference to my paper from Jeffrey Shallit, Jan 23 2015

Moved a comment to A056857 where it belonged. - N. J. A. Sloane, May 02 2015.