The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A350081 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A350081

Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).
(history; published version)

#34 by Michel Marcus at Thu Feb 17 14:08:13 EST 2022

STATUS

reviewed

approved

#33 by Omar E. Pol at Thu Feb 17 13:50:03 EST 2022

STATUS

proposed

reviewed

#32 by Steven Finch at Thu Feb 17 13:49:23 EST 2022

STATUS

editing

proposed

#31 by Steven Finch at Thu Feb 17 13:43:34 EST 2022

LINKS

Steven Finch, <a href="http://arxiv.org/abs/2202.07621">Second best, Third worst, Fourth in line</a>, arxiv:2202.07621 [math.CO], 2022.

CROSSREFS

Cf. A001865, A350078, A350079, A350080, A350275, A350276.

STATUS

approved

editing

#30 by Andrew Howroyd at Tue Dec 28 14:11:02 EST 2021

STATUS

reviewed

approved

#29 by Michel Marcus at Tue Dec 28 13:37:08 EST 2021

STATUS

proposed

reviewed

#28 by Jean-François Alcover at Tue Dec 28 09:20:08 EST 2021

STATUS

editing

proposed

#27 by Jean-François Alcover at Tue Dec 28 09:20:03 EST 2021

MATHEMATICA

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];

b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[3]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 3]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];

T[n_] := With[{p = b[n, {Infinity, Infinity, Infinity}]}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];

Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

STATUS

approved

editing

#26 by Alois P. Heinz at Sun Dec 19 07:24:18 EST 2021

STATUS

editing

approved

#25 by Alois P. Heinz at Sun Dec 19 07:24:03 EST 2021

MAPLE

b:= proc(n, i, l) option remember; `if`(n=0, x^subs(infinity=0, l)[3],

`if`(i<1, 0, add(b(n-i*j, i-1, , sort([l[], i$j])[1..3])*g(i)*binomial(n-1, i-1), i=1..n))

g(i)^j*combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i)))

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, , [infinity$3])):

STATUS

approved

editing

Discussion

Sun Dec 19

07:24

Alois P. Heinz: shortened ...